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 Yes b Binomial n 15 p 1 5 0 20 What is the expected number of questions that Bart would get right E X n p 15 0 20 3 c What is the probability that Bart answers exactly 3 questions correctly 15 P X 3 0 20 3 0 80 12 0 2501389 3 OR P X 3 CDF 3 CDF 2 0 648 0 398 0 250 d What is the probability that Bart would get at most 5 of the questions right P X 5 CDF 5 0 939 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 P X 8 1 CDF 7 1 0 996 0 004 f Find the probability that Bart answers between 4 and 6 including both 4 and 6 questions correctly P 4 X 6 CDF 6 CDF 3 0 982 0 648 0 334 OR 15 15 15 P 4 X 6 0 20 4 0 80 11 0 20 5 0 80 10 0 20 6 0 80 9 4 5 6 0 18760417 0 10318229 0 04299262 0 33377908 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 0 00000095 12 0 99999994 13 14 0 00000006 0 00000000 13 14 1 00000000 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 X number of sales Binomial n 5 p 0 30 5 P X 2 0 30 2 0 70 3 0 3087 2 OR P X 2 CDF 2 CDF 1 0 837 0 528 0 309 2 Often On time Parcel Service OOPS delivers a package to the wrong address with probability 0 05 on any delivery Suppose that each delivery is independent of all the others There were 7 packages delivered on a particular day a What is the probability that at least one of them was delivered to the wrong address P X 1 1 CDF 0 1 0 698 0 302 b What is the probability that exactly two of them were delivered to the wrong address 7 P X 2 0 05 2 0 95 5 0 0406235 2 OR P X 2 CDF 2 CDF 1 0 996 0 956 0 040 c What is the probability that at most two of them were delivered to the wrong address P X 2 CDF 2 0 996 d What is the probability that at least two of them were delivered to the wrong address P X 2 1 CDF 1 1 0 956 0 044 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 F F F F S 0 70 0 70 0 70 0 70 0 30 0 07203 Geometric Distribution X the number of independent trials until the first success Then P X x 1 p x 1 p 1 E X 4 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 F F F S 0 85 0 85 0 85 0 15 0 092 Geometric p 0 15 b What is the probability that your first reward occurs on your seventh trial F F F F F F S 0 85 0 85 0 85 0 85 0 85 0 85 0 15 0 0566 Geometric p 0 15 c What is the probability that your get three rewards in ten trials Binomial n 10 p 0 15 10 P X 3 0 15 3 0 85 7 0 1298 3 d What is the probability that your third reward occurs on your tenth trial 9 trials 2 S s 7 F s S 9 0 15 2 0 85 7 0 15 0 03895 2 Negative Binomial k 3 p 0 15 Negative Binomial Distribution X the number of independent trials until the k th success Then x 1 p k 1 p x k P X x k 1 E X EXCEL e k p x k k 1 k 2 V X NEGBINOMDIST x k k p k 1 p p2 gives P X x What is the probability that your fourth reward occurs on your fifteenth trial 14 trials 3 S s 11 F s S 14 0 15 3 0 85 11 0 15 0 030837 3 Negative Binomial k 4 p 0 15 f What is the probability that your get four rewards in fifteen trials Binomial n 15 p 0 15 15 P X 4 0 15 4 0 85 11 0 11564 4 Let X be a random variable with a Geometric distribution with probability of success p Then P X a 1 p k 1 p 1 p a p 1 1 p k a 1 1 p a a 0 1 2 3 OR X number of independent attempts needed to get the first success P X a P the first a attempts are failures 1 p a Ex a 0 1 2 3 Let X denote the number of rolls of a fair 6 seded die needed to observe the first …
View Full Document
Unlocking...