Binomial Probability Distribution A binomial probability distribution results from a procedure where: 1) There are a fixed number of trials 2) The trials are independent* 3) Each trial has only two possible outcomes 4) The probabilities are the same for all trials *when sampling without replacement, if the sample size is less than 5% of the population size, we can treat the events as if they were independent

Binomial Probability Distribution Example: Flip 100 coins and count the number of heads: There are 100 trials (100 flips of a coin) The outcome of one flip doesn’t affect the next flip Each trial has two outcomes – heads or tails The probability of heads is always ½

Notation We usually call the two outcomes success and failure p probability of success in one trial q = 1-p probability of failure in one trial n number of trials x specific number of successes in n trials (can be 0 to n) P(x) probability of exactly x successes in the n trials

Example What’s the probability of randomly guessing 8 questions right on a 10 question multiple choice test, where each question has 4 possible answers.   n = 10 x = 8 p = ¼ = 0.25 q = 1 - ¼ = 0.75 We’re looking for P(8)

The Formula

Example What’s the probability of randomly guessing 8 questions right on a 10 question multiple choice test, where each question has 4 possible answers. n = 10 x = 8 p = ¼ = 0.25 q = 0.75 How would we find the P(at least 8 right)? P(at least 8) = P(8) + P(9) + P(10) =0.000416

You try A company makes widgets, with a 2% defect rate. If you sample and test 15 widgets, what is the probability that: Exactly one has a defect? None have defects? At least one has a defect?

Homework 4.3: 1, 5, 7, 17, 19, 21, 25, 29, 33