Chapter 5 - Probability And Counting Rules Iii (1)
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PROBABILITY
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY A numerical measure of the likelihood that a specific event will occur. An event that cannot occur has zero probability which is called an impossible event
and if an event that is certain to occur has a probability equal to 1 is called sure event. There are four (4) basic probability rules that will be helpful in solving probability problems. 1. The probability of an event is within the range 0 to 1.
2. The sum of the probabilities of all simple events for an experiment is always 1.
For an experiment: 3. If an event cannot occur, its probability is 0. 4. If an event is certain, then the probability is 1. "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Complimentary Events The complement of an event is the set of outcomes in the sample space that are not included in
the outcome of event . The complement of is denoted by (read as E prime).
The rule for complimentary events are denoted by,
(Formula 5-9) (Formula 5-9) (Formula 5-9)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Three Conceptual Approaches to Probability A. Classical Probability
Classical probability assumes that all outcomes in the sample space are equally likely to occur.
(Formula 5-10)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities a.
Getting king of hearts
b.
Getting a spade
c.
Getting a 5 or a clubs
d.
Getting a 5 or a 7
e.
Getting a card which is not a spade
f.
Getting 14 of clubs
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting king of hearts
Solution
Since there is only one king of hearts in an event E and 52 possible outcomes in the sample space.
Therefore, the probability of getting king of hearts is 0.02 or 2%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a spade
Solution
There are 13 spades so there are 13 outcomes in an event E.
Therefore, the probability of getting a spade is 0.25 or 25%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a 5 or a clubs
Solution There are four 5s and 13 clubs in an event E, but the 5 of spades are counted twice in this listing. Thus, there are 16 possible outcomes of drawing 5 or a clubs. Therefore, the probability of getting a 5 or a clubs is 0.31 or 31%. This is an example of the inclusive or.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a 5 or a 7
Solution There are four 5s and four 7s in an event E. Therefore, the probability of getting a 5 or a 7 is 0.15 or 15%. This is an example of the exclusive or.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a card which is not a spade
Solution
There are 39 cards which is not a spade in an event E.
Therefore, the probability of getting a card which is not a spade is 0.75 or 75%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a card which is not a spade
Alternative Solution Recall that P(spade) is 0.25 or 25%, we simply deduct this to 1 to obtain the probability of getting a non-spade card. We’ll use P(E’) = 1 – P(E), where P(E’) is the probability of getting a non-spade card. Therefore, the probability of getting a card which is not a spade is 0.75 or 75%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a 14 of clubs
Solution It is impossible to get a 14 of clubs in the sample space of an ordinary deck of card. Therefore, the probability of getting a 14 of clubs is 0%. This is an example of impossible event.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Three Conceptual Approaches to Probability B. Empirical or Relative Frequency Probability
Empirical probability is the type of probability that uses frequency distribution based on observations to determine numerical probabilities of events.
(Formula 5-11)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example In a sample of 50 college students, 18 are
freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities:
Year Level
Frequency
Freshman
18
Sophomore
23
A student is a freshman or a sophomore
Junior
2
A student is neither a freshman nor a junior
Senior
7
Total
50
A student is a freshman
A student is not a senior
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Solution
PROBABILITY
Example In a sample of 50 college students, 18 are
freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities: A student is a freshman A student is a freshman or a sophomore A student is neither a freshman nor a junior A student is not a senior
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Solution
To obtain the probability of selecting a
freshman we simply divide the number of freshmen by the sample space. Therefore, the probability is 0.36 or 36%.
PROBABILITY
Example In a sample of 50 college students, 18 are
freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities: A student is a freshman A student is a freshman or a sophomore A student is neither a freshman nor a junior A student is not a senior
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Solution
We need to add the frequency of the two
year level (or classes)
Therefore, the probability is 0.82 or 82%.
PROBABILITY
Example In a sample of 50 college students, 18 are
freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities:
Solution
Note that neither a freshman nor a junior
means that the student is either a sophomore or a senior.
A student is a freshman A student is a freshman or a sophomore A student is neither a freshman nor a junior A student is not a senior
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Therefore, the probability is 0.60 or 60%.
PROBABILITY
Example In a sample of 50 college students, 18 are
freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities:
Solution
In order to find the probability of not a
senior, we need to subtract the probability of senior from 1.
A student is a freshman A student is a freshman or a sophomore A student is neither a freshman nor a junior A student is not a senior
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Therefore, the probability is 0.86 or 86%.
PROBABILITY
Three Conceptual Approaches to Probability C. Subjective Probability
Subjective probability is the probability assigned to an event based on subjective judgment, experience, information, and belief.
For example,
A sportswriter may say that there is 90% probability that University of the East Red Warriors will win the UAAP championships.
A physician may say that, on the basis of his diagnosis, there is a 60% chance that the patient will recover.
A financial analysis may say that there is 80% probability that peso dollar exchange rate will decrease by 3 pesos.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
THE ADDITION RULE AND MULTIPLICATION RULE PROBABILITY
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE A lot of problems involve determine the probability of two or more events. This is when
independent, dependent, and mutually exclusive comes into the picture in dealing with probability. There are important things to note about mutually exclusive, independent, and
dependent events. The first is of which is mutually exclusive are always dependent Secondly is independent events are never mutually exclusive Lastly is dependent events may or may not be mutually exclusive. The exception of the first and second is that when at least one of the two events has a zero
probability.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Independent, Dependent, Mutually Exclusive Events Two events A and B are independent events if the fact that A occurs not affect the probability of B
occurring. In other words, A and B are independent events if, or
Two events A and B are dependent events for which the outcome or occurrence of event A affects the outcome or occurrence of event B in such a way that the probability is changed. In other words, A and B are dependent events if, or Two events A and B are mutually exclusive events if they cannot occur at the same time.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Addition Rules for Probability Rule 1
When two events A and B are mutually
exclusive, the probability that A or B will occur is
(Formula 5-12)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Addition Rules for Probability
Rule 2
If A and B are not mutually exclusive, then
(Formula 5-13)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A box contains 4 red marbles, 8 blue marbles, and 7 green marbles. If a person selects a marble at
random, find the probability that’s either a red or green marble.
Solution Since the box contains 4 red marbles, 7 green marbles, and a total of 19 marbles. Therefore, selecting a marble at random that’s either a red or green is 0.58 or 58%. The events are mutually exclusive.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A single card is drawn from an ordinary deck of card. Find the probability that it is a queen or a
diamonds.
Solution Since a queen of diamonds means a queen and a diamond, it has been counted twice-once as a queen and once as a diamond; thus, one of the outcomes must be deducted, as shown Therefore, drawing a queen or a diamond from an ordinary deck is 0.31 or 31%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example In a certain insurance company there are 20
senior salespersons and 30 junior salespersons; 8 senior and 14 junior salespersons are males. If a salesperson is selected, find the probability that the salesperson is a senior or a female.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
The sample space
Salespers on
Male
Female
Total
Senior
18
12
20
Junior
14
16
30
Total
22
28
50
Solution
ADDITION RULE AND MULTIPLICATION RULE
Example In a certain insurance company there are 20
senior salespersons and 30 junior salespersons; 8 senior and 14 junior salespersons are males. If a salesperson is selected, find the probability that the salesperson is a senior or a female.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Solution
The probability is
0.72 or 72%.
ADDITION RULE AND MULTIPLICATION RULE
The multiplication rules can be applied to determine the probability of two or more events
that occur in sequence. The probability of the intersection of two events is called their joint probability. It is written as .
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE Rule 1 When two events are independent, the probability of both occurring is
(Formula 5-14)
Rule 2 When two events are dependent, the probability of both occurring is
(Formula 5-15)
Rule 3 When two events are mutually exclusive their joint probability is always zero. If A and B are two
mutually exclusive events, then (Formula 5-16) "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A die is rolled and a coin is flipped. Find the probability of getting a 5 on the die and tail on the coin.
Solution Using the sample space for the die which is 1, 2, 3, 4, 5, 6; for the coin which is Head, Tail. Therefore, the probability of getting a 5 on the die and tail on the coin is 0.08 or 8%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A box contains 3 red balls, 8 blue balls, and 9 green balls. A first ball is selected, and then it is
replaced. A second ball is selected. Find the probability of selecting 2 red balls
1 blue ball and 1 green ball Solution 2 red balls Therefore, selecting 2 red balls is 0.02 or 2%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A box contains 3 red balls, 8 blue balls, and 9 green balls. A first ball is selected, and then it is
replaced. A second ball is selected. Find the probability of selecting 2 red balls
1 blue ball and 1 green ball Solution 1 blue ball and 1 green ball Therefore, selecting 1 blue and 1 green ball is 0.18 or 18%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A SJS survey found that one out of 5 Filipinos say they are in favor of the death penalty for heinous
crimes. If the people are selected at random, find the probability that all three will say that they are in favor of death penalty.
Solution Let D denote that a person is in favor of death penalty. Then Therefore, the probability that all three will say that they are in favor of death penalty is 0.01 or
1%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example Reina owns a collection of 25 bags, of which 6 are made by Guess. If the 2 bags are selected at
random, find the probability that both are made by Guess.
Solution Since the events are dependent, Therefore, the probability that both are made by Guess is 0.05 or 5%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example The RSS Financing Inc. found that 50% of the members had salary long (S) with the financing
company. Of these members 8% also had a calamity loan (C). If a member is selected at random find the probability that the member has both loans with the company.
Solution
Note that the events are dependent, Therefore, the probability that the member has both loans with the company is 0.04 or 4%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
MARGINAL AND CONDITIONAL PROBABILITIES PROBABILITY
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
MARGINAL AND CONDITIONAL PROBABILITIES
Marginal Probability Marginal Probability is a probability of a single event without consideration of any other event; it is
also called single probability. It can be computed using the formula
(Formula 5-17) Where
are k mutually exclusive and collectively exhaustive events.
Recall that two events are mutually exclusive if both the events cannot occur simultaneous, while
collectively exhaustive if one of the events must occur.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
MARGINAL AND CONDITIONAL PROBABILITIES
Conditional Probability Conditional probability is probability that an event will occur given that another event has already
occurred. If A and B are two events, then the conditional probability is given as P(A|B) and reads as “the probability of A given that B has already occurred.” In symbol,
(Formula 5-18) and given that and
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
MARGINAL AND CONDITIONAL PROBABILITIES Example A box contains blue and red balls. A person select two balls without replacement. If the probability
of selecting a blue ball and a red ball is 12/30, and the probability of selecting a blue ball on the first draw is 3/5, find the probability of selecting a red ball on the second draw, given that the first ball selected was a blue ball.
Solution Let
B = selecting a blue ball R = selecting a red ball
Then, Thus, the probability of selecting a red ball on the second draw given that the first ball selected was
blue is 0.67 or 67%. "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
MARGINAL AND CONDITIONAL PROBABILITIES Example In a fast-food chain, 75% of the customers orders chicken meal. If 40% of the customers orders
chicken meal and sundae, find the probability that the customer orders chicken meal will also order a sundae.
Solution Let
C = the customer orders chicken meal S = the customer orders sundae
Then, Thus, the customer has a 0.53 or 53% probability of ordering sundae, given that he/she ordered
chicken meal first. "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION PROBABILITY
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Number Number of of TVs TVs owned owned
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION
Frequency Frequency
Relative Relative Frequency Frequency
Suppose the above table shows the frequency and relative frequency distribution of the number of TV owned by 500 families residing in the City of Manila.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION Suppose one family is randomly selected
from this population. The process of random selecting a family is called a random or chance experiment.
Number of TVs owned
Let X denote the number of TVs owned by
the selected family. Then X can assume any of the 4 possible values (0, 1, 2, and 3) recorded in the leftmost column of the table. The value assumed by X depends on the family is selected. Hence, this value depends on the outcome
of a random experiment. Therefore, X is referred to random variable. "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Frequency
Relative Frequency
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION
A random variable is a function or rule that assigns a number to each outcome of an
experiment, it is called chance variable. In general, a random variable is denoted by X. A random variable can be discrete or continuous. A discrete random variable assumes values that can be counted, while A continuous random variable can assume all values between any two specific values; A variable obtained by measuring, or Contained one or more intervals
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION
A discrete probability distribution consists of the values a random variable can assume
and the corresponding probabilities of the values. The probabilities are determined theoretically or by observation. There are several requirements for a distribution of a discrete random variable. For a discrete random variable X than can assume values , 1. , for all (The probability outcome is between 0 and 1). 2.
(The sum of all possible outcomes is 1.0).
3. The listing is exhaustive (All possible outcomes are included). 4. The outcomes are mutually exclusive (The outcomes cannot occur at the same time).
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION
Example Construct a probability distribution for rolling
a die.
Solution Since the sample spaces of a die is 1, 2, 3, 4, 5, 6 and each outcome has a probability of , the distribution is,
Outcome X Probability P(X) Cumulative F(X)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION Event Example Construct a probability distribution for
tossing three coins. Let X represent the number of tails.
Solution The 8 possible events, and the
corresponding values for X, are:
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
TTT
3
TTH
2
THT
2
HTT
2
THH
1
HTH
1
HHT
1
HHH
0
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION
Example Construct a probability distribution for
tossing three coins. Let X represent the number of tails.
Solution Therefore, the probability distribution for the
number of tails occurring in three coin tosses is:
Outcome X Probability P(X) Cumulative F(X)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION PROBABILITY
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION A binomial experiment is one that
possesses the following properties: The experiment consists of n repeated trials; Each trial results in an outcome that may be
classified as a success or a failure (hence the name, binomial); The probability of a success, denoted by p,
remains constant from trial to trial and repeated trials are independent.
The number of successes X in n trials of a
binomial experiment is called a binomial random variable. "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
The probability distribution of the random variable X is called a binomial distribution, and is given by the formula:
Formula (5-19) Where:
BINOMIAL PROBABILITY DISTRIBUTION
Examples of binomial experiments Tossing a coin 20 times to see how many
tails occur. Asking 200 people if they watch ABC news. Rolling a die to see if a 5 appears.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Examples which aren't binomial experiments
Rolling a die until a 6 appears (not a fixed
number of trials) Asking 20 people how old they are (not two
outcomes) Drawing 5 cards from a deck for a poker
hand (done without replacement, so not independent)
BINOMIAL PROBABILITY DISTRIBUTION
Example What is the probability of rolling exactly two sixes in 6 rolls of a die?
Solution There are five things you need to do to work a binomial story problem.
1.
Define Success first. Success must be for a single trial.
2.
Define the probability of success (p):
3.
Find the probability of failure:
4.
Define the number of trials:
5.
Define the number of successes out of those trials:
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION Example What is the probability of rolling exactly two
sixes in 6 rolls of a die?
Solution Anytime a six appears, it is a success
(denoted S) and anytime something else appears, it is a failure (denoted F). The ways you can get exactly 2 successes in 6 trials are given below. The probability of each is written to the right of the way it could occur. Because the trials are independent, the probability of the event (all six dice) is the product of each probability of each outcome (die):
1.
FFFFSS 5/6 * 5/6 * 5/6 * 5/6 * 1/6 * 1/6 = (1/6)^2 * (5/6)^4
2.
FFFSFS 5/6 * 5/6 * 5/6 * 1/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
3.
FFFSSF 5/6 * 5/6 * 5/6 * 1/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
4.
FFSFFS 5/6 * 5/6 * 1/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
5.
FFSFSF 5/6 * 5/6 * 1/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
6.
FFSSFF 5/6 * 5/6 * 1/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
7.
FSFFFS 5/6 * 1/6 * 5/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION Example What is the probability of rolling exactly two
sixes in 6 rolls of a die?
Solution Anytime a six appears, it is a success
(denoted S) and anytime something else appears, it is a failure (denoted F). The ways you can get exactly 2 successes in 6 trials are given below. The probability of each is written to the right of the way it could occur. Because the trials are independent, the probability of the event (all six dice) is the product of each probability of each outcome (die):
8.
FSFFSF 5/6 * 1/6 * 5/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
9.
FSFSFF 5/6 * 1/6 * 5/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
10.
FSSFFF 5/6 * 1/6 * 1/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
11.
SFFFFS 1/6 * 5/6 * 5/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
12.
SFFFSF 1/6 * 5/6 * 5/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
13.
SFFSFF 1/6 * 5/6 * 5/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
14.
SFSFFF 1/6 * 5/6 * 1/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
15.
SSFFFF 1/6 * 1/6 * 5/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION
Example What is the probability of rolling exactly two sixes in 6 rolls of a die?
Solution Notice that each of the 15 probabilities are exactly the same: (1/6)^2 * (5/6)^4. Also, note that the 1/6 is the probability of success and you needed 2 successes. The 5/6 is the
probability of failure, and if 2 of the 6 trials were success, then 4 of the 6 must be failures. Note that 2 is the value of x and 4 is the value of n-x. Further note that there are fifteen ways this can occur. This is the number of ways 2 successes can
be occur in 6 trials without repetition and order not being important, or a combination of 6 things, 2 at a time.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION Example A coin is tossed 10 times. What is the
probability that exactly 6 heads will occur?
Using formula 5-19,
Solution
Therefore, the probability that exactly 6
heads will occur is 0.21 or 21%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION PROBABILITY
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION
A Poisson distribution is the probability distribution that results from a Poisson experiment. Attributes of a Poisson Experiment A Poisson experiment is a statistical experiment that has the following properties: The experiment results in outcomes that can be classified as successes or failures. The average number of successes (μ) that occurs in a specified region is known. The probability that a success will occur is proportional to the size of the region. The probability that a success will occur in an extremely small region is virtually zero. Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a
period of time, etc.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION
Notation The following notation is helpful, when we talk about the Poisson distribution.
– A constant equal to approximately 2.71828. (Actually, e is the base of the natural logarithm system.)
– The mean number of successes that occur in a specified region.
– The actual number of successes that occur in a specified region.
– The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is μ.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION Poisson Distribution A Poisson random variable is the number
of successes that result from a Poisson experiment. The probability distribution of a Poisson
random variable is called a Poisson distribution. The Poisson distribution has the following
properties:
Given the mean number of successes (μ)
that occur in a specified region, we can compute the Poisson probability based on the following formula: Poisson Formula. Suppose we conduct a
Poisson experiment, in which the average number of successes within a given region isμ. Then, the Poisson probability is:
(Formula 5-20) Where
The mean of the distribution is equal to μ . The variance is also equal to μ .
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
.
POISSON PROBABILITY DISTRIBUTION
Example The average number of homes sold by the Acme Realty company is 2 homes per day. What is the
probability that exactly 3 homes will be sold tomorrow?
Solution This is a Poisson experiment in which we know the following: ; since 2 homes are sold per day, on average. ; since we want to find the likelihood that 3 homes will be sold tomorrow. ; since e is a constant equal to approximately 2.71828.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION
Example The average number of homes sold by the Acme Realty company is 2 homes per day. What is the
probability that exactly 3 homes will be sold tomorrow?
Solution We plug these values into the formula 5-20 as follows: Thus, the probability of selling 3 homes tomorrow is 0.18 or 18%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION Example If there are 500 customers per eight-hour day in a check-out lane, what is the probability that there
will be exactly 3 in line during any five-minute period?
Solution The expected value during any one five minute period would be 500 / 96 = 5.21.
The 96 is because there are 96 five-minute periods in eight hours. So, you expect about 5.2 customers in 5
minutes and want to know the probability of getting exactly 3.
This is a Poisson experiment in which we know the following: 5.21; since 5.2 customers in 5 minutes, on average. ; since we want to find the likelihood that 3 will be in line during any five-minute period. ; since e is a constant equal to approximately 2.71828.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION
Example If there are 500 customers per eight-hour day in a check-out lane, what is the probability that there
will be exactly 3 in line during any five-minute period?
Solution We plug these values into the formula 5-20 as follows: Thus, the probability that there will be exactly 3 in line during any five-minute period is
0.14 or 14%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY A numerical measure of the likelihood that a specific event will occur. An event that cannot occur has zero probability which is called an impossible event
and if an event that is certain to occur has a probability equal to 1 is called sure event. There are four (4) basic probability rules that will be helpful in solving probability problems. 1. The probability of an event is within the range 0 to 1.
2. The sum of the probabilities of all simple events for an experiment is always 1.
For an experiment: 3. If an event cannot occur, its probability is 0. 4. If an event is certain, then the probability is 1. "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Complimentary Events The complement of an event is the set of outcomes in the sample space that are not included in
the outcome of event . The complement of is denoted by (read as E prime).
The rule for complimentary events are denoted by,
(Formula 5-9) (Formula 5-9) (Formula 5-9)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Three Conceptual Approaches to Probability A. Classical Probability
Classical probability assumes that all outcomes in the sample space are equally likely to occur.
(Formula 5-10)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities a.
Getting king of hearts
b.
Getting a spade
c.
Getting a 5 or a clubs
d.
Getting a 5 or a 7
e.
Getting a card which is not a spade
f.
Getting 14 of clubs
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting king of hearts
Solution
Since there is only one king of hearts in an event E and 52 possible outcomes in the sample space.
Therefore, the probability of getting king of hearts is 0.02 or 2%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a spade
Solution
There are 13 spades so there are 13 outcomes in an event E.
Therefore, the probability of getting a spade is 0.25 or 25%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a 5 or a clubs
Solution There are four 5s and 13 clubs in an event E, but the 5 of spades are counted twice in this listing. Thus, there are 16 possible outcomes of drawing 5 or a clubs. Therefore, the probability of getting a 5 or a clubs is 0.31 or 31%. This is an example of the inclusive or.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a 5 or a 7
Solution There are four 5s and four 7s in an event E. Therefore, the probability of getting a 5 or a 7 is 0.15 or 15%. This is an example of the exclusive or.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a card which is not a spade
Solution
There are 39 cards which is not a spade in an event E.
Therefore, the probability of getting a card which is not a spade is 0.75 or 75%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a card which is not a spade
Alternative Solution Recall that P(spade) is 0.25 or 25%, we simply deduct this to 1 to obtain the probability of getting a non-spade card. We’ll use P(E’) = 1 – P(E), where P(E’) is the probability of getting a non-spade card. Therefore, the probability of getting a card which is not a spade is 0.75 or 75%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example A card is drawn from an ordinary deck of card. Find these probabilities Getting a 14 of clubs
Solution It is impossible to get a 14 of clubs in the sample space of an ordinary deck of card. Therefore, the probability of getting a 14 of clubs is 0%. This is an example of impossible event.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Three Conceptual Approaches to Probability B. Empirical or Relative Frequency Probability
Empirical probability is the type of probability that uses frequency distribution based on observations to determine numerical probabilities of events.
(Formula 5-11)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
PROBABILITY
Example In a sample of 50 college students, 18 are
freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities:
Year Level
Frequency
Freshman
18
Sophomore
23
A student is a freshman or a sophomore
Junior
2
A student is neither a freshman nor a junior
Senior
7
Total
50
A student is a freshman
A student is not a senior
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Solution
PROBABILITY
Example In a sample of 50 college students, 18 are
freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities: A student is a freshman A student is a freshman or a sophomore A student is neither a freshman nor a junior A student is not a senior
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Solution
To obtain the probability of selecting a
freshman we simply divide the number of freshmen by the sample space. Therefore, the probability is 0.36 or 36%.
PROBABILITY
Example In a sample of 50 college students, 18 are
freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities: A student is a freshman A student is a freshman or a sophomore A student is neither a freshman nor a junior A student is not a senior
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Solution
We need to add the frequency of the two
year level (or classes)
Therefore, the probability is 0.82 or 82%.
PROBABILITY
Example In a sample of 50 college students, 18 are
freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities:
Solution
Note that neither a freshman nor a junior
means that the student is either a sophomore or a senior.
A student is a freshman A student is a freshman or a sophomore A student is neither a freshman nor a junior A student is not a senior
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Therefore, the probability is 0.60 or 60%.
PROBABILITY
Example In a sample of 50 college students, 18 are
freshmen, 23 are sophomore, 2 are junior, and 7 are senior. Set up a frequency distribution and find the following probabilities:
Solution
In order to find the probability of not a
senior, we need to subtract the probability of senior from 1.
A student is a freshman A student is a freshman or a sophomore A student is neither a freshman nor a junior A student is not a senior
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Therefore, the probability is 0.86 or 86%.
PROBABILITY
Three Conceptual Approaches to Probability C. Subjective Probability
Subjective probability is the probability assigned to an event based on subjective judgment, experience, information, and belief.
For example,
A sportswriter may say that there is 90% probability that University of the East Red Warriors will win the UAAP championships.
A physician may say that, on the basis of his diagnosis, there is a 60% chance that the patient will recover.
A financial analysis may say that there is 80% probability that peso dollar exchange rate will decrease by 3 pesos.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
THE ADDITION RULE AND MULTIPLICATION RULE PROBABILITY
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE A lot of problems involve determine the probability of two or more events. This is when
independent, dependent, and mutually exclusive comes into the picture in dealing with probability. There are important things to note about mutually exclusive, independent, and
dependent events. The first is of which is mutually exclusive are always dependent Secondly is independent events are never mutually exclusive Lastly is dependent events may or may not be mutually exclusive. The exception of the first and second is that when at least one of the two events has a zero
probability.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Independent, Dependent, Mutually Exclusive Events Two events A and B are independent events if the fact that A occurs not affect the probability of B
occurring. In other words, A and B are independent events if, or
Two events A and B are dependent events for which the outcome or occurrence of event A affects the outcome or occurrence of event B in such a way that the probability is changed. In other words, A and B are dependent events if, or Two events A and B are mutually exclusive events if they cannot occur at the same time.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Addition Rules for Probability Rule 1
When two events A and B are mutually
exclusive, the probability that A or B will occur is
(Formula 5-12)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Addition Rules for Probability
Rule 2
If A and B are not mutually exclusive, then
(Formula 5-13)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A box contains 4 red marbles, 8 blue marbles, and 7 green marbles. If a person selects a marble at
random, find the probability that’s either a red or green marble.
Solution Since the box contains 4 red marbles, 7 green marbles, and a total of 19 marbles. Therefore, selecting a marble at random that’s either a red or green is 0.58 or 58%. The events are mutually exclusive.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A single card is drawn from an ordinary deck of card. Find the probability that it is a queen or a
diamonds.
Solution Since a queen of diamonds means a queen and a diamond, it has been counted twice-once as a queen and once as a diamond; thus, one of the outcomes must be deducted, as shown Therefore, drawing a queen or a diamond from an ordinary deck is 0.31 or 31%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example In a certain insurance company there are 20
senior salespersons and 30 junior salespersons; 8 senior and 14 junior salespersons are males. If a salesperson is selected, find the probability that the salesperson is a senior or a female.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
The sample space
Salespers on
Male
Female
Total
Senior
18
12
20
Junior
14
16
30
Total
22
28
50
Solution
ADDITION RULE AND MULTIPLICATION RULE
Example In a certain insurance company there are 20
senior salespersons and 30 junior salespersons; 8 senior and 14 junior salespersons are males. If a salesperson is selected, find the probability that the salesperson is a senior or a female.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Solution
The probability is
0.72 or 72%.
ADDITION RULE AND MULTIPLICATION RULE
The multiplication rules can be applied to determine the probability of two or more events
that occur in sequence. The probability of the intersection of two events is called their joint probability. It is written as .
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE Rule 1 When two events are independent, the probability of both occurring is
(Formula 5-14)
Rule 2 When two events are dependent, the probability of both occurring is
(Formula 5-15)
Rule 3 When two events are mutually exclusive their joint probability is always zero. If A and B are two
mutually exclusive events, then (Formula 5-16) "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A die is rolled and a coin is flipped. Find the probability of getting a 5 on the die and tail on the coin.
Solution Using the sample space for the die which is 1, 2, 3, 4, 5, 6; for the coin which is Head, Tail. Therefore, the probability of getting a 5 on the die and tail on the coin is 0.08 or 8%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A box contains 3 red balls, 8 blue balls, and 9 green balls. A first ball is selected, and then it is
replaced. A second ball is selected. Find the probability of selecting 2 red balls
1 blue ball and 1 green ball Solution 2 red balls Therefore, selecting 2 red balls is 0.02 or 2%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A box contains 3 red balls, 8 blue balls, and 9 green balls. A first ball is selected, and then it is
replaced. A second ball is selected. Find the probability of selecting 2 red balls
1 blue ball and 1 green ball Solution 1 blue ball and 1 green ball Therefore, selecting 1 blue and 1 green ball is 0.18 or 18%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example A SJS survey found that one out of 5 Filipinos say they are in favor of the death penalty for heinous
crimes. If the people are selected at random, find the probability that all three will say that they are in favor of death penalty.
Solution Let D denote that a person is in favor of death penalty. Then Therefore, the probability that all three will say that they are in favor of death penalty is 0.01 or
1%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example Reina owns a collection of 25 bags, of which 6 are made by Guess. If the 2 bags are selected at
random, find the probability that both are made by Guess.
Solution Since the events are dependent, Therefore, the probability that both are made by Guess is 0.05 or 5%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
ADDITION RULE AND MULTIPLICATION RULE
Example The RSS Financing Inc. found that 50% of the members had salary long (S) with the financing
company. Of these members 8% also had a calamity loan (C). If a member is selected at random find the probability that the member has both loans with the company.
Solution
Note that the events are dependent, Therefore, the probability that the member has both loans with the company is 0.04 or 4%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
MARGINAL AND CONDITIONAL PROBABILITIES PROBABILITY
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
MARGINAL AND CONDITIONAL PROBABILITIES
Marginal Probability Marginal Probability is a probability of a single event without consideration of any other event; it is
also called single probability. It can be computed using the formula
(Formula 5-17) Where
are k mutually exclusive and collectively exhaustive events.
Recall that two events are mutually exclusive if both the events cannot occur simultaneous, while
collectively exhaustive if one of the events must occur.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
MARGINAL AND CONDITIONAL PROBABILITIES
Conditional Probability Conditional probability is probability that an event will occur given that another event has already
occurred. If A and B are two events, then the conditional probability is given as P(A|B) and reads as “the probability of A given that B has already occurred.” In symbol,
(Formula 5-18) and given that and
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
MARGINAL AND CONDITIONAL PROBABILITIES Example A box contains blue and red balls. A person select two balls without replacement. If the probability
of selecting a blue ball and a red ball is 12/30, and the probability of selecting a blue ball on the first draw is 3/5, find the probability of selecting a red ball on the second draw, given that the first ball selected was a blue ball.
Solution Let
B = selecting a blue ball R = selecting a red ball
Then, Thus, the probability of selecting a red ball on the second draw given that the first ball selected was
blue is 0.67 or 67%. "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
MARGINAL AND CONDITIONAL PROBABILITIES Example In a fast-food chain, 75% of the customers orders chicken meal. If 40% of the customers orders
chicken meal and sundae, find the probability that the customer orders chicken meal will also order a sundae.
Solution Let
C = the customer orders chicken meal S = the customer orders sundae
Then, Thus, the customer has a 0.53 or 53% probability of ordering sundae, given that he/she ordered
chicken meal first. "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION PROBABILITY
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Number Number of of TVs TVs owned owned
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION
Frequency Frequency
Relative Relative Frequency Frequency
Suppose the above table shows the frequency and relative frequency distribution of the number of TV owned by 500 families residing in the City of Manila.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION Suppose one family is randomly selected
from this population. The process of random selecting a family is called a random or chance experiment.
Number of TVs owned
Let X denote the number of TVs owned by
the selected family. Then X can assume any of the 4 possible values (0, 1, 2, and 3) recorded in the leftmost column of the table. The value assumed by X depends on the family is selected. Hence, this value depends on the outcome
of a random experiment. Therefore, X is referred to random variable. "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Frequency
Relative Frequency
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION
A random variable is a function or rule that assigns a number to each outcome of an
experiment, it is called chance variable. In general, a random variable is denoted by X. A random variable can be discrete or continuous. A discrete random variable assumes values that can be counted, while A continuous random variable can assume all values between any two specific values; A variable obtained by measuring, or Contained one or more intervals
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION
A discrete probability distribution consists of the values a random variable can assume
and the corresponding probabilities of the values. The probabilities are determined theoretically or by observation. There are several requirements for a distribution of a discrete random variable. For a discrete random variable X than can assume values , 1. , for all (The probability outcome is between 0 and 1). 2.
(The sum of all possible outcomes is 1.0).
3. The listing is exhaustive (All possible outcomes are included). 4. The outcomes are mutually exclusive (The outcomes cannot occur at the same time).
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION
Example Construct a probability distribution for rolling
a die.
Solution Since the sample spaces of a die is 1, 2, 3, 4, 5, 6 and each outcome has a probability of , the distribution is,
Outcome X Probability P(X) Cumulative F(X)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION Event Example Construct a probability distribution for
tossing three coins. Let X represent the number of tails.
Solution The 8 possible events, and the
corresponding values for X, are:
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
TTT
3
TTH
2
THT
2
HTT
2
THH
1
HTH
1
HHT
1
HHH
0
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTION
Example Construct a probability distribution for
tossing three coins. Let X represent the number of tails.
Solution Therefore, the probability distribution for the
number of tails occurring in three coin tosses is:
Outcome X Probability P(X) Cumulative F(X)
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION PROBABILITY
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION A binomial experiment is one that
possesses the following properties: The experiment consists of n repeated trials; Each trial results in an outcome that may be
classified as a success or a failure (hence the name, binomial); The probability of a success, denoted by p,
remains constant from trial to trial and repeated trials are independent.
The number of successes X in n trials of a
binomial experiment is called a binomial random variable. "PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
The probability distribution of the random variable X is called a binomial distribution, and is given by the formula:
Formula (5-19) Where:
BINOMIAL PROBABILITY DISTRIBUTION
Examples of binomial experiments Tossing a coin 20 times to see how many
tails occur. Asking 200 people if they watch ABC news. Rolling a die to see if a 5 appears.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
Examples which aren't binomial experiments
Rolling a die until a 6 appears (not a fixed
number of trials) Asking 20 people how old they are (not two
outcomes) Drawing 5 cards from a deck for a poker
hand (done without replacement, so not independent)
BINOMIAL PROBABILITY DISTRIBUTION
Example What is the probability of rolling exactly two sixes in 6 rolls of a die?
Solution There are five things you need to do to work a binomial story problem.
1.
Define Success first. Success must be for a single trial.
2.
Define the probability of success (p):
3.
Find the probability of failure:
4.
Define the number of trials:
5.
Define the number of successes out of those trials:
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION Example What is the probability of rolling exactly two
sixes in 6 rolls of a die?
Solution Anytime a six appears, it is a success
(denoted S) and anytime something else appears, it is a failure (denoted F). The ways you can get exactly 2 successes in 6 trials are given below. The probability of each is written to the right of the way it could occur. Because the trials are independent, the probability of the event (all six dice) is the product of each probability of each outcome (die):
1.
FFFFSS 5/6 * 5/6 * 5/6 * 5/6 * 1/6 * 1/6 = (1/6)^2 * (5/6)^4
2.
FFFSFS 5/6 * 5/6 * 5/6 * 1/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
3.
FFFSSF 5/6 * 5/6 * 5/6 * 1/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
4.
FFSFFS 5/6 * 5/6 * 1/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
5.
FFSFSF 5/6 * 5/6 * 1/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
6.
FFSSFF 5/6 * 5/6 * 1/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
7.
FSFFFS 5/6 * 1/6 * 5/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION Example What is the probability of rolling exactly two
sixes in 6 rolls of a die?
Solution Anytime a six appears, it is a success
(denoted S) and anytime something else appears, it is a failure (denoted F). The ways you can get exactly 2 successes in 6 trials are given below. The probability of each is written to the right of the way it could occur. Because the trials are independent, the probability of the event (all six dice) is the product of each probability of each outcome (die):
8.
FSFFSF 5/6 * 1/6 * 5/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
9.
FSFSFF 5/6 * 1/6 * 5/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
10.
FSSFFF 5/6 * 1/6 * 1/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
11.
SFFFFS 1/6 * 5/6 * 5/6 * 5/6 * 5/6 * 1/6 = (1/6)^2 * (5/6)^4
12.
SFFFSF 1/6 * 5/6 * 5/6 * 5/6 * 1/6 * 5/6 = (1/6)^2 * (5/6)^4
13.
SFFSFF 1/6 * 5/6 * 5/6 * 1/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
14.
SFSFFF 1/6 * 5/6 * 1/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
15.
SSFFFF 1/6 * 1/6 * 5/6 * 5/6 * 5/6 * 5/6 = (1/6)^2 * (5/6)^4
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION
Example What is the probability of rolling exactly two sixes in 6 rolls of a die?
Solution Notice that each of the 15 probabilities are exactly the same: (1/6)^2 * (5/6)^4. Also, note that the 1/6 is the probability of success and you needed 2 successes. The 5/6 is the
probability of failure, and if 2 of the 6 trials were success, then 4 of the 6 must be failures. Note that 2 is the value of x and 4 is the value of n-x. Further note that there are fifteen ways this can occur. This is the number of ways 2 successes can
be occur in 6 trials without repetition and order not being important, or a combination of 6 things, 2 at a time.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
BINOMIAL PROBABILITY DISTRIBUTION Example A coin is tossed 10 times. What is the
probability that exactly 6 heads will occur?
Using formula 5-19,
Solution
Therefore, the probability that exactly 6
heads will occur is 0.21 or 21%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION PROBABILITY
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION
A Poisson distribution is the probability distribution that results from a Poisson experiment. Attributes of a Poisson Experiment A Poisson experiment is a statistical experiment that has the following properties: The experiment results in outcomes that can be classified as successes or failures. The average number of successes (μ) that occurs in a specified region is known. The probability that a success will occur is proportional to the size of the region. The probability that a success will occur in an extremely small region is virtually zero. Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a
period of time, etc.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION
Notation The following notation is helpful, when we talk about the Poisson distribution.
– A constant equal to approximately 2.71828. (Actually, e is the base of the natural logarithm system.)
– The mean number of successes that occur in a specified region.
– The actual number of successes that occur in a specified region.
– The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is μ.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION Poisson Distribution A Poisson random variable is the number
of successes that result from a Poisson experiment. The probability distribution of a Poisson
random variable is called a Poisson distribution. The Poisson distribution has the following
properties:
Given the mean number of successes (μ)
that occur in a specified region, we can compute the Poisson probability based on the following formula: Poisson Formula. Suppose we conduct a
Poisson experiment, in which the average number of successes within a given region isμ. Then, the Poisson probability is:
(Formula 5-20) Where
The mean of the distribution is equal to μ . The variance is also equal to μ .
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
.
POISSON PROBABILITY DISTRIBUTION
Example The average number of homes sold by the Acme Realty company is 2 homes per day. What is the
probability that exactly 3 homes will be sold tomorrow?
Solution This is a Poisson experiment in which we know the following: ; since 2 homes are sold per day, on average. ; since we want to find the likelihood that 3 homes will be sold tomorrow. ; since e is a constant equal to approximately 2.71828.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION
Example The average number of homes sold by the Acme Realty company is 2 homes per day. What is the
probability that exactly 3 homes will be sold tomorrow?
Solution We plug these values into the formula 5-20 as follows: Thus, the probability of selling 3 homes tomorrow is 0.18 or 18%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION Example If there are 500 customers per eight-hour day in a check-out lane, what is the probability that there
will be exactly 3 in line during any five-minute period?
Solution The expected value during any one five minute period would be 500 / 96 = 5.21.
The 96 is because there are 96 five-minute periods in eight hours. So, you expect about 5.2 customers in 5
minutes and want to know the probability of getting exactly 3.
This is a Poisson experiment in which we know the following: 5.21; since 5.2 customers in 5 minutes, on average. ; since we want to find the likelihood that 3 will be in line during any five-minute period. ; since e is a constant equal to approximately 2.71828.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
POISSON PROBABILITY DISTRIBUTION
Example If there are 500 customers per eight-hour day in a check-out lane, what is the probability that there
will be exactly 3 in line during any five-minute period?
Solution We plug these values into the formula 5-20 as follows: Thus, the probability that there will be exactly 3 in line during any five-minute period is
0.14 or 14%.
"PROBABILITY THEORY IS NOTHING BUT COMMON SENSE REDUCED TO CALCULATION." - P.S. LAPLACE
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