STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 2 4 2 5 Binomial Distribution 1 The number of trials n is fixed 2 Each trial has two possible outcomes success and failure 3 The probability of success p is the same from trial to trial 4 The trials are independent 5 X number of successes in n independent trials Then n P X k p k 1 p n k k n k k n Ck p 1 p where k 0 1 n E X n p Var X n p 1 p SD X n p 1 p 1 Bart Simpson takes a multiple choice exam in his Statistics 101 class The exam has 15 questions each has 5 possible answers only one of which is correct Bart did not study for the exam so he guesses independently on every question Let X denote the number of questions that Bart gets right a Is it appropriate to use Binomial model for this problem b What is the expected number of questions that Bart would get right c What is the probability that Bart answers exactly 3 questions correctly d What is the probability that Bart would get at most 5 of the questions right e What is the probability that Bart would get more than half of the questions right i e what is the probability that Bart would get at least 8 of the questions right f Find the probability that Bart answers between 4 and 6 including both 4 and 6 questions correctly Binomial n 15 p 0 20 EXCEL Outcome Probability k CDF at k 0 1 0 03518437 0 13194140 0 1 0 03518437 0 16712577 2 0 23089744 2 0 39802321 3 0 25013890 3 0 64816210 4 0 18760417 4 0 83576628 5 0 10318229 5 0 93894857 6 0 04299262 6 0 98194119 7 0 01381906 7 0 99576025 8 0 00345476 8 0 99921501 9 0 00067176 9 0 99988677 10 0 00010076 10 0 99998754 11 0 00001145 11 0 99999899 12 13 0 00000095 0 00000006 12 13 0 99999994 1 00000000 14 0 00000000 14 1 00000000 15 0 00000000 BINOMDIST k n p 0 BINOMDIST k n p 1 2 An automobile salesman thinks that the probability of making a sale is 0 30 If he talks to five customers on a particular day what is the probability that he will make exactly 2 sales Assume independence 3 A major oil company has decided to drill independent test wells in the Alaskan wilderness The probability of any well producing oil is 0 30 Find the probability that the fifth well is the first to produce oil Geometric Distribution X the number of independent trials until the first success Then P X x 1 p x 1 p E X 4 1 p x 1 2 3 Var X 1 p p2 A slot machine at a casino randomly rewards 15 of the attempts Assume that all attempts are independent a What is the probability that your first reward occurs on your fourth trial b What is the probability that your first reward occurs on your seventh trial c What is the probability that your get three rewards in ten trials d What is the probability that your third reward occurs on your tenth trial Negative Binomial Distribution X the number of independent trials until the k th success Then x 1 x k k P X x p 1 p k 1 E X EXCEL k p x k k 1 k 2 V X NEGBINOMDIST x k k p k 1 p p2 gives P X x e What is the probability that your fourth reward occurs on your fifteenth trial f What is the probability that your get four rewards in fifteen trials
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