Binomial Probability Distribution

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Binomial Probability Distribution Consider the experiment in which a die is rolled three times. On each roll, the probability of getting a six (S) is 1/6 and the probability of getting a non-six (N) is 5/6. The tree diagram below summarizes the results of the experiment. S N 1/6 5/6 1/6 5/6 1/6 5/6 1/6 5/6 1/6 5/6 1/6 5/6 1/6 5/6 Now you fill in all of the other probabilities. What is the p(3sixes)? What is p(2 sixes)?

Binomial Probability Distribution Outcomes: 3 sixes 2sixes,1 non-six 1 six,2 non-six 0 sixes Probability: Compare to Let’s define variables: p = probability of a successful outcome, q = probability of a failure, and p + q = 1

Binomial Probability Distribution BPD may be used whenever the following criteria are met. *The experiment consists of 2 possible outcomes. (p and q) *The experiment has a fixed number of trials. (repeated n times) *The trials are independent. *The probability of success, p, is the same in each trial.

Ex 1: A multiple choice test contains 20 questions Ex 1: A multiple choice test contains 20 questions. Each question has five choices for the correct answer. Only one of the choices is correct. What is the probability of making an 80% B- with random guessing? n=20 r=16 p=.20 q=.80 P(16 correct) = = 4845(6.5536x10^-12)(.4096) WOW!!!

Ex2: In a survey of 1000 students in a school, 950 indicated that they were right-handed. Find the probability that AT LEAST one of four randomly chosen students from the school are LEFT-HANDED. Technically speaking, this is not a binomial experiment because the sampling of the four students is done without replacement. In this experiment the probability of the 1st student being right-handed is 950/1000=.95. The probability that the 2nd student is right-handed is either 949/999 or 950/999 depending on whether the 1st student was right-handed. These two probabilities (as well as those for the 3rd and 4th student’s being right-handed) are so close to .95 that we will use the binomial probability theorem even though the theorem requires a constant probability. using the Complement P(at least 1 lefty) = 1-P(all right-handed) =1-.815 =.185 or 18.5%