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MATH 218 ASSIGNMENT
DUE TUESDAY, FEB 28, 2006
Problems which are labeled “Haskell” are from the book, “Problem Sets for Math 218,” by Cymra Haskell (which used to be one of the required texts for the course, but can still be found online). You won’t need that book to look the problems up, since they’re reproduced here—but you will need that book if you want to see the solutions worked-out.
Problem 1. Compute the following binomial coefficients. Do this manually (i.e. without using the ‘binomial coefficient’ button on your calculator). You may use the calculator to do the final multiplications.
(a)
(b)
(c)
(d)
Problem 2. When Robyn Hud fires an arrow at a target, he has the following probabilities of scoring points:
Points Probability 10 0. 9 0. 8 0.
He fires three arrows at a target, with the shots independent of each other. Thus he can score anywhere from 24 through 30 points.
(a) Draw a tree illustrating the situation. (b) Using the method of generating functions, find the probability that he scores 24 points; 25 points;... ; through 30 points.
Problem 3. How many four-digit numbers are there (i.e. numbers formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) if
(a) Leading 0’s are allowed, and repetitions are allowed? (b) Leading 0’s are allowed, but repetitions are not allowed?
Answer the same questions if leading 0’s are not allowed.
Problem 4. An urn contains two white balls and three black balls. Each of three women, Ann, Betty, and Charmin, in that order, draws one ball at random from the urn and does not replace it. The first to draw a white ball earns $10. 1
2 MATH 218 ASSIGNMENT DUE TUESDAY, FEB 28, 2006
(a) Determine the expectation of each of the players. (b) Determine the expectation of each player if they do replace the ball after drawing it.
(It may help to draw a tree.)
Problem 5. (Haskell, Sheet II.1) A committee consisting of five people is to be chosen from a group of 20 people.
(a) How many different committees are possible? (b) How many committees are there that include both Aaron and Bernadette?
Problem 6. (Haskell, Sheet II.2) (Adapted from an exercise in “An Introduction to Probability and Its Applications” by Larsen and Marx.) A woman is attacked by two assailants. A day later the police show her a line-up of seven suspects. The line-up includes the two guilty people.
(a) How many ways can the woman choose two people out of the line-up? (b) How many ways can the woman choose two people out of the line-up exactly one of whom is guilty?
Problem 7. (Haskell, Sheet II.17) (Adapted from an exercise in “An Introduction to Probability and Its Applications” by Larsen and Marx.) A woman is attacked by two assailants. A day later the police show her a line-up of seven suspects. The line-up includes the two assailants. Suppose the woman never saw her assailants and chooses two people from the line-up completely at random.
(a) What is the probability that among the two people she chooses exactly one of them is guilty? (b) What is the probability that neither of the people she chooses is guilty?
Problem 8. (Haskell, Sheet II.19) Widgets coming off an assembly line have a 10% defective rate. Five widgets are selected at random and tested.
(a) What is the probability that all five of them are defective? (b) What is the probability that the first two are defective and the other three are not? (c) What is the probability that the second and fifth ones are defective and the other three are not? (d) How many different outcomes result in exactly two of the five being defective? (e) What is the probability that exactly two of the five are defective?
Problem 9. (Adapted from Haskell, Sheet II.22) Suppose that, when properly trained, people have an 80% chance of passing a lie detector test even when lying. Eight members of a terrorist group that were involved in a bombing are detained by the police and given a lie detector test. What is the probability that:
(a) None of the terrorists will be detected lying? (b) Exactly one of the terrorists will be detected lying? (c) One or more of the terrorists will be detected lying?
Problem 10. (Haskell, Sheet II.16) Consider a population consisting of 20 women and 10 men.
(a) How many samples of size 3 are there? (b) How many samples of size 3 are there that consist of 2 women and 1 man? (c) If a simple random sample of size three is selected from this population, what is the proba- bility that it will consist of 2 women and 1 man?
