1. MATHPOWER TM 12, WESTERN EDITION Chapter 9 Probability Distributions 9.1 9.1.1

2. 9.1.2 Using the Binomial Theorem to Calculate Probabilities What is the probability of correctly guessing the outcome of exactly one out of four rolls of a die? The probability of correctly guessing one roll of the die is. The probability of incorrectly guessing the outcome is. The probability of one correct and three incorrect guesses is. The correct guess can occur in any one of the four rolls so there are ways of arranging the correct guess. P(one correct guess in four rolls) = = NOTE: This experiment is called a binomial experiment because it has two outcomes:

3. 9.1.3 Using the Binomial Theorem to Calculate Probabilities For this experiment, let p represent the probability of a correct guess and q represent the probability of an incorrect guess. Use the binomial theorem. Expand and evaluate (p + q) 6, where Thus, the probability of correctly guessing the outcome of six out of six rolls is Find the probability of correctly guessing the outcome of exactly six out of six rolls of the die. NOTE: This is a binomial experiment because there are only two outcomes for each roll: guessing correctly or incorrectly.

4. 9.1.4 Using the Binomial Theorem to Calculate Probabilities [cont’d] You can use the table feature of a graphing calculator to calculate probabilities. Using the Binomial Probability Distribution feature of the TI-83: DISTR 0: binompdf binompdf (number of trials, probability of success, x-value) n = p = P( ) binompdf( =

5. 9.1.5 Binomial Distribution Binomial distribution occurs when two outcomes are possible in an experiment: success or failure Binomial distribution is a function, P(x). The binomial distribution depends on two quantities: the probability of success for one outcome: p the number of trials in the experiment: n The sum of all the values of P(x) is 1. For every value of x, 0 ≤ P(x) ≤ 1.

6. 9.1.8 Binomial Distribution - Applications The binomial distribution is the pattern of probabilities for the outcomes of repeated independent and identical trials. If p is the probability of success and q is the probability of failure (q = 1 - p), then the probability of x successes in n trials is: 1. Hockey cards, chosen at random from a set of 20, are given away inside cereal boxes. Stan needs one more card to complete his set so he buys five boxes of cereal. What is the probability that he will complete his set? P(x successes) = n C x p x q n - x The probability of Stan completing his set is

7. 2. Seven coins are tossed. What is the probability of four tails and three heads? P(x successes) = n C x p x q n - x = 3. A true-false test has 12 questions. Suppose you guess all 12. What is the probability of exactly seven correct answers? P(x successes) = n C x p x q n - x = The probability of seven correct answers is The probability of four heads and three tails is 9.1.9 Binomial Distribution - Applications n = (number of trials) x = (number of successes) p = (probability of success) q = (probability of failure) P( successes) = n = (number of trials) x = (number of successes) p = (probability of success) q = (probability of failure) P( successes) =

8. 9.1.10 4. A test consists of 10 multiple choice questions, each with four possible answers. To pass the test, one must answer at least nine questions correctly. Find the probability of passing, if one were to guess the answer for each question. P(x successes) = n C x p x q n - x P(x successes) = P( successes) + P( successes) The probability of passing is Binomial Distribution - Applications

9. 9.1.11 5.A family has nine children. What is the probability that there is at least one girl? This can be best solved using the compliment, that is, the probability of zero girls: P(x successes) = n C x p x q n - x = The probability of zero girls is, therefore the probability of at least one girl is Binomial Distribution - Applications P( successes) = n = (number of trials) x = (number of successes) p = (probability of success) n = (probability of failure)

10. 9.1.13 Suggested Questions: Pages 407 and 408 1, 3, 6, 8-10, 14, 16, 17, 18 ac, 19 a, 21 a