CHAPTER 5

DISCRETE PROBABILITY DISTRIBUTIONS

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5.3 THE BINOMIAL PROBABILITY DISTRIBUTION

Suppose we want to find the probability that head will turn up 6 times when a coin is tossed 10 times. The binomial probability distribution is best suited for this type of question. To use the binomial probability distribution, the experiment must meet the following conditions:1. There are n identical trials. In other words, the experiment is repeated

n times and each repetition is performed under identical conditions. Note that each repetition of the experiment is called a trial or a Bernoulli trial.

2. Each trial has only two possible outcomes. That is, the outcome is either a “success” or a “failure.”

3. The probability of each outcome remains constant. The probability of success is denoted by p and probability of failure is denoted by q. Therefore, p + q = 1.

4. The trials are independent. In other words, the occurrence of one trial does not affect the probability of the other.

Note:a. Success refers to the outcome to which the question is stated. Then

failure refers to the outcome that the question does not stated.b. Success does not mean that the outcome is favorable. Likewise,

failure does not mean that the outcome is unfavorable.2

The Binomial Probability Distribution

Which is the following are binomial experiment? Explain why.a. Rolling a die many times and observing the number of spots.b. Rolling a die many times and observing whether the number obtained is even or odd?c. Selecting a few voters from a very large population of voters and observing whether or not each of them favors a certain proposition in an election when 54% of all voters are known to be in favor of this proposition.

Example #14

Solution

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The Binomial Probability DistributionSolution

b. Rolling a die many times and observing whether the number obtained is even or odd

c. Selecting a few voters from a very large population of voters and observing whether or not each of them favors a certain proposition in an election when 54% of all voters are known to be in favor of this proposition

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The Binomial Probability Distribution and Binomial Formula

So, knowing that an experiment is a binomial experiment, then we can

apply the Binomial probability distribution to find the probability of

exactly x successes in n trials by using the binomial formula

where n = total number of trials p = probability of success q = 1 – p = probability of failurex = number of successes in n trialsn - x = number of failures in n trials

( ) x n xn xP x C p q

number of ways to obtain x success from n trialsn xC

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!!( )!n x

nC

x n x

The Binomial Probability Distribution and Binomial Formula

According to a Harris Interactive poll, 52% of American college graduates have Facebook accounts. Suppose that this result is true for the current population of American college graduates.

a. Let x be a binomial random variable that denotes the number of American college graduates in a random sample of 15 who have Facebook accounts. What are the possible values that x can assume?

b. Find the probability that exactly 9 American college graduates in a sample of 15 have Facebook accounts.

Example #15 Solution

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The Binomial Probability Distribution and Binomial Formula

According to an October 27, article in Newsweek, 65% of Americans said they take expired medicines. Suppose this result is true of the current population of Americans. Find the probability that the number of Americans in a random sample of 22 who take expired medicines is

a. exactly 17 b. none c. exactly 9

Example #16

Solution

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Using the Table of Binomial Probabilities

So, rather than calculate the probability of x success in n trials by using the Binomial formula, we can use

Table I-A of Appendix A – Tables of Binomial Probabilities. Lets do this by solving the following problem.

According to a March 25, 2007 Pittsburgh Post-Gazette article, 30% to 40% of U.S. taxpayers cheat on their returns. Suppose that 30% of all current U.S. taxpayers cheat on their returns. Use the binomial probabilities table (Table I of Appendix C) or technology to find the probability that the number of U.S. taxpayers in a random sample of 14 who cheat on their taxes is

a. at least 8 b. at most 3 c. 3 to 7

Example #17

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Solution

Probability of Success and the Shape of the Binomial Distribution

The shape of a binomial distribution depends on the value of the probability of success. If p = 0.50, then the binomial probability distribution is symmetric.

If p < 0.50, then binomial probability distribution is right skewed.

If p > 0.50, then binomial probability distribution is left skewed.

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5.4 Mean and Standard Deviation of the Binomial Distribution

For a binomial distribution, we can still use the formulas in Sections 5.3 and 5.4 to compute its mean and standard deviation.

However a simpler methods are:

where,n = total number of trialsp = probability of successq = probability of failure.

npqnp

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µ = E(x) = Σ x P(x) 2 2( )x P x

Mean and Standard Deviation of the Binomial Distribution

A fast food chain store conducted a taste survey before marketing a new hamburger. The results of the survey showed that 70% of the people who tried hamburger liked it. Encouraged by this result, the company decided to market the new hamburger. Assume that 70% of all people like this hamburger. On a certain day, eight customers bought it for the first time.

a. Let x denote the number of customers in this sample of eight who will like the hamburger. Using the binomial probabilities table, obtain the probability distribution of x and draw a graph of the probability distribution. Determine the mean and standard deviation of x.

a. Using the probability distribution of part a, find the probability that exactly three of the eight customers will like this hamburger.

Example #18

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Mean and Standard Deviation of the Binomial Distribution

a. Using Table I of Appendix A, the probability distribution of x is,

Solution

x P(x)

0 0.0001

1 0.0012

2 0.0100

3 0.0467

4 0.1361

5 0.2541

6 0.2965

7 0.1977

8 0.0576

Bar Graph for the probability distribution

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 1 2 3 4 5 6 7 8

x

P(x

)

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13

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FACTORIALS, COMBINATIONS, AND PERMUTATIONS

Factorial is denoted by the symbol “!”. The factorial of a number is calculated by multiplying all integers from the number to 1.

Formal DefinitionThe symbol n!, is define as the product of all the integers from n to 1. In other words,

n! = n(n - 1)(n – 2)(n – 3) · · · 3 · 2 · 1

Also note that by definition,0! = 1

Example #9

3! 3 2 1 6(9 3)! 6! 6 5 4 3 2 1 7209! 9 8 7 6 5 4 3 2 1 362,880

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CombinationsSuppose we want to select two books from 6 books, and the following conditions applies:1. This type of selection is without replacement.2. The order in which selection is made is not important. That is, it is the same selection whether we select book 1 first (1,2) or book 2 first (2,1).Then, the question is how many ways can we make this selection? Of course, we can do it the old fashion way, write the selection as,

(1,2) (1,3) (1,4) (1,5) (1,6)(2,3) (2,4) (2,5) (2,6)(3,4) (3,5) (3,6)(4,5) (4,6)(5,6)

Therefore, we can select 2 books from 6 books 15 different ways.Note:Each of the possible selection is called a combination. 16

CombinationsDefinitionCombinations give the number of ways x element can be selected from n distinct elements. The total number of combinations is given by,

and is read as “the number of combinations of n elements selected x at a time.”

The formula for the number of combinations for selecting x from n distinct elements is,

Note:

n xC

!!( )!n x

nC

x n x

! ! !

!( )! ! 0!n n

n n nC

n n n n

!n1 0

! ! !0!( 0)! 0! !n

n n nC

n n

!n117

Combinations

Example #10

2

5 3

7 4

4 0

3 3

5! 5! 5 3 2 14 103!(5 3)! 3! 2! 3 2 1 2 1

7! 7! 7 6 5 4 3 2 135

4!(7 4)! 4! 3! 4 3 2 1 3 2 1

1

1

C

C

C

C

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PermutationsPermutation is similar to combination except that the order of selection is important. For example, how many ways can we select two letters from “A”, “B”, and “C” without replacement and if the order does not matter?

This selection is a combination and the total number of combinations is

These 3 ways are “AB”, “AC, and “BC”. Now suppose the order matters, then the two letter-combination can be selected or arranged in 6 ways:

“AB”, “BA”, “AC”, “CA”, “BC” and “CB”

These 6 selections are called 6 permutations or arrangements.

3 2

3!3 ways.

2!1!C

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Permutations

DefinitionPermutations give the total selection of x elements from n different elements such that the order of selection matter. The notation and formula is !

( )!n x

nP

n x

Example #11

6 2

8 4

6! 6 5 4 !6 5 30

4! 4 !

6! 8 7 6 5 4 !8 7 6 5 1,680

4! 4 !

P

P

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Permutations

A ski patrol unit has nine members available for duty, and two of them are to be sent to rescue an injured skier. In how many ways can two of these nine members be selected? Now suppose the order of selection is important. How many arrangements are possible in this case?

Example #12

4

9 2

9, 2

9! 9 7!8a.2!7!

n x

C

2 7!

9 2

9 4 36

9! 9 8 7!b.

7!P

7!9 8 72

Solution

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Permutations

An environmental agency will randomly select 4 houses from a block containing 25 houses for a radon check. How many total selections are possible? How many permutations are possible?

Example #13

25 4

25, 4

25! 25 24a.

4!21!

n x

C 23 22 21!

4 3 2 21!

25 4

25 23 22 12,650

25! 25 24 23 22 21!b.

21!P

21!303,600

Solution

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