1 The Binomial Probability Distribution - Faculty Web Pages [PDF]

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The Binomial Probability Distribution

De…nition 1 A probability model has the binomial distribution if 1. there are two possible outcomes (success and failure) and the probability of success, p, is constant; 2. there are a …xed number of trials, n; 3. the trials are independent. Exercise 2 An experiment consists of ‡ipping a fair coin 8 times and counting the number of tails. Is this experiment a binomial probability distribution?

Exercise 3 A multiple choice test contains 20 questions. Each question has …ve choices for the correct answer. Only one of the choices is correct. With random guessing, is this a binomial probability distribution?

Exercise 4 A multiple choice test contains 20 questions. Each question has four or …ve choices for the correct answer. Only one of the choices is correct. With random guessing is this a binomial probability distribution?

Exercise 5 A box contains 3 white balls, 4 red balls and 5 black balls. Three balls are randomly selected with replacement. Is counting the number of red balls a binomial probability distribution?

Exercise 6 A box contains 3 white balls, 4 red balls and 5 black balls. Three balls are randomly selected without replacement. Is counting the number of red balls a binomial probability distribution?

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Knowing an experiment has a binomial probability distribution provides some nice bene…ts. 1. The mean of a binomial distribution is

= np;

2. The standard deviation of a binomial distribution is

p np (1

p);

n k

3. The probability of exactly k successes is nk pk (1 p) . This value is easily computed on the TI-83/84 with the binompdf(n; p; k) command. 4. The cumulative probability of 0; 1; 2; 3; :::; or k successes is

k X

n i

pi (1

n i

p)

i=0

This value is easily computed on the TI-83/84 with the binomcdf(n; p; k) command. Problem 7 An experiment consists of ‡ipping a fair coin 8 times and counting the number of tails. 1. Find the probability of seeing exactly 3 tails.

2. Find the probability of seeing at most 3 tails.

3. Find the probability of seeing at least 3 tails.

4. Find the probability of seeing between 5 and 7 tails.

5. Would it be unusual to see no more than 2 tails?

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Problem 8 A true/false choice test contains 20 questions. 1. What is the average number of corrects answers with random guessing?

2. What is the standard deviation for corrects answers with random guessing?

3. What is the probability of making an 80 with random guessing?

4. What is the probability of making at least an 80 with random guessing?

5. What is the probability of scoring between 60 and 70 (inclusive) with random guessing?

6. Would it be unusual to score a 65 with random guessing?

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Exercises 1. Section 5.2 Navidi/Monk: 5-7, 17-21,27-32

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