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Statistics / Elementary Statistics: Picturing the World 6 / Chapter 4.2 / Problem 17E

Elementary Statistics: Picturing the World | 6th Edition | ISBN: 9780321911216 | Authors: Ron Larson; Betsy Farber

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Finding Binomial Probabilities In Exercise,

Chapter 4, Problem 17E

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QUESTION:

Problem 17E

Finding Binomial Probabilities In Exercise, find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B to find the probabilities.

Ease of Voting Twenty-seven percent of likely U.S. voters think that it is too easy to vote in the United States. You randomly select 12 likely U.S. voters. Find the probability that the number of likely U.S. voters who think that it is too easy to vote in the United States is (a) exactly three, (b) at least four, and (c) less than eight.

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QUESTION:

Problem 17E

Finding Binomial Probabilities In Exercise, find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B to find the probabilities.

Ease of Voting Twenty-seven percent of likely U.S. voters think that it is too easy to vote in the United States. You randomly select 12 likely U.S. voters. Find the probability that the number of likely U.S. voters who think that it is too easy to vote in the United States is (a) exactly three, (b) at least four, and (c) less than eight.

ANSWER:

Solution 17E

Step1 of 4:

From the given problem we have an experiment, In that 27% of U.S. voters think that it is too easy to vote in the United States. Also, we have n = 12.

Here our goal is:

a). We need to find the probability of getting exactly three U.S. voters think that it is too easy to vote in the United States.

b). We need to find the probability of getting at least four U.S. voters think that it is too easy to vote in the United States.

c). We need to find the probability of getting less than eight U.S. voters think that it is too easy to vote in the United States.

Step2 of 4:

a).

Let us consider a random variable ‘x’ it presents the number of U.S. voters think that it is too easy to vote in the United States. Let us assume that ‘x’ follows the Binomial distribution with parameters ‘n and p’. That is:

$X\sim{}B(n,\ p)$

$X\sim{}B(12,\ 0.27)$

And we know that the probability mass function of the Binomial distribution is:

$P(X)={(}_{x}^{n})\ {p}^{x}\ {(1-p)}^{n\ -\ x},$ $for\ x=0,1,2,...,n.$

Then, The probability of getting exactly three U.S. voters think that it is too easy to vote in the United States is:

$P(X=3)={(}_{3}^{12})\ {(0.27)}^{3}\ {(1-0.27)}^{12\ -\ 3}$

$=220(0.0196)(0.0588)$

$=0.2538$

Therefore, P(X = 3) = 0.2538.

Step3 of 4:

b).

The probability of getting at least four U.S. voters think that it is too easy to vote in the United States is:

$P(X\geq{}4)=1-P(X<4)$

$=1-\{P(X=0)+P(X=1)+P(X=2)+P(X=3)\}$

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