Discrete Math Section 16.3 Use the Binomial Probability theorem to find the probability of a given outcome on repeated independent trials. Flip a coin three times - each flip is independent of all the previous flips Outcomes: HHH HHT HTH HTT THH THT TTH TTT Groups probability 1 3H 1/8 3 2H 1T 3/8 3 1H 2T 3/8 1 3T 1/8 Pascal’s Triangle

Binomial Probability Theorem Suppose an experiment consists of a sequence of n repeated independent trials, each trial having two possible outcomes, A or not A. If on each trial P(A) = p and P(not A) = 1-p, then the binomial expansion [ p + (1-p)] n gives the probabilities for all occurrences of A. n C n p n + n C n-1 p n-1 (1-p) 1 + n C n-2 p n-2 (1-p) 2 … n C 0 (1-p) n

Examples A coin is tossed 8 times. What is the probability of getting exactly 5 heads and 3 tails? note: two outcomes…heads or tails repeated independent trials P(H) = p = ½ P(T) = 1-p = ½ 8 C 5 p 5 (1-p) 3 8 C 5 (1/2) 5 (1/2) 3 = 7/32

example A quiz consists of 10 multiple choice questions. Each question has three possible answers. If you guess at each question, what is the probability of getting at least eight correct answers?

example A die is rolled four times. What is the probability of getting exactly three fives?

Assignment Page 616 Problems 2-12 even, 15,17