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Assignment 2 Due 16 December 2009, Wednesday Please hand in to the Resources Office Binomial Experiment Experiment of m repeated trials where the following are valid: 1. all trials are statistically independent (in the sense that knowing the outcome of any particular one of them does not change one’s assessment of chance related to any others); 2. each trial results in only one of two possible outcomes, labeled as “success” and “failure”; 3. the probability of success on each trial, denoted by p, remains constant. Binomial Random Variable X = Number of trials that result in a success X is said to have Binomial distribution with parameters m and p. Binomial PMF p (x) = m x m x p (1 p) x where m m! x x!(m x)! Factorial m! = m x (m – 1) x …. x 2 x 1 0! = 1 1! = 1 2! = 2 x 1 = 2 3! = 3 x 2 x 1 = 6 4! = 4 x 3 x 2 x 1 = 24 Expectation E(X) = m p Variance Var (X) = m p (1 – p) Example 1 Two building contractors, A and B, have competed for 50 contracts. A won 20 and B won 30. The contractors asked to tender for 3 new contracts. What is the probability that A) Contractor A will win all the contracts (0.064); B) Contractor B will win at least one contract (0.936); C) Contractor A will win exactly two contracts (0.288)? Example 2 A food-packaging apparatus under-fills 10% of the containers. Find the probability that for any particular 5 containers the number of under-filled will be: A) exactly 3 (0.0081); B) exactly 2 (0.0729); C) zero (0.590); D) at least 1 (0.41). Example 3 A particular jury consists of 7 jurors. Each juror has a 0.2 chance of making the wrong decision, independently of the others. If the jury reaches a decision by majority rule, what is the probability that it will reach a wrong decision? (0.0333) Example 4 In a shotgun seminar, the topic is announced 1 week in advance, but the speaker is randomly chosen when the seminar begins. Suppose that you attend 12 seminars with 19 other participants. (i) What is the expected number of times you will be selected as speaker? (0.6) (ii) What is the probability that you will be selected exactly twice? (0.099) (iii) Suppose that the 20 participants are divided into five teams of four for preparation. What is the probability that your team will be selected exactly twice? (0.283)