2. 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 4 C 1 ways of arranging the correct guess. P(one correct guess in four rolls) = = 0.386 NOTE: This experiment is called a binomial experiment because it has two outcomes: guessing correctly AND guessing incorrectly

3. 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 P(x successes) = n C x p x q n - x

4. Binomial Distribution - Applications 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? = 0.2 The probability of Stan completing his set is 20%.

5. 2. Seven coins are tossed. What is the probability of four tails and three heads? = 0.273 3. A true-false test has 12 questions. Suppose you guess all 12. What is the probability of exactly seven correct answers? = 0.193 The probability of seven correct answers is 19%. The probability of four heads and three tails is 27%. Binomial Distribution - Applications n = 7 (number of trials) x = 4 (number of successes) p = (probability of success) q = (probability of failure) P(4 successes) = n = 12 (number of trials) x = 7 (number of successes) p = (probability of success) q = (probability of failure) P(7 successes) =

6. 4. A family has nine children. What is the probability that there is at least one girl? This can be best solved using the compliment. 1 – P(NO GIRLS) number of trials The probability of zero girls is 0.00195, therefore the probability of at least one girl is 1 - 0.00195 = 0.998. Binomial Distribution - Applications 1 – (1/2) 9

7. 5. 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) = P(9 successes) + P(10 successes) = 0.000 0296 The probability of passing is 0.003%. Binomial Distribution - Applications So this means that you need to find the probability of getting 9 OR 10 right.

8. Binomial Distribution - Applications 6.While pitching for the Chicago White Sox, 4 of every 7 pitches Chris Sale threw in the first 5 innings were strikes. What is the probability that 3 of the next 4 pitches will be strikes? n = 4 (number of pitches) x = 3 (number of strikes) p = (probability of success) q = (probability of failure) P(x successes) = n C x p x q n - x = 0.32 P(3 successes) = Let’s assume that there are only two possible outcomes, strikes or balls.

9. Using the Binomial Theorem to Calculate Probabilities [cont’d] Using the Binomial Probability Distribution feature of the TI-83: DISTR binompdf binompdf (number of trials, probability of success, x-value) n = 4 x = 3 p = 4/7 binompdf(4, 4/7, 3) = 0.32

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