Counting

1. A closet contains 6 diﬀerent pairs of shoes. Five shoes are drawn at random. What is the probability that at least one pair of shoes is obtained? 2. At a camera factory, an inspector checks 20 cameras and ﬁnds that three of them need adjustment before they can be shipped. Another employee carelessly mixes the cameras up so that no one knows which is which. Thus, the inspector must recheck the cameras one at a time until he locates all the bad ones. (a) What is the probability that no more than 17 cameras need to be rechecked? (b) What is the probability that exactly 17 must be rechecked? 3. We consider permutations of the string ”ABACADAFAG”. How many permutations are there? How many of them don’t have any A next to other A? How many of them have at least two A’s next to each other? 4. A monkey is typing random numerical strings of length 7 using the digits 1 through 9 (not 0). Call the digits 1, 2, and 3 ”lows”, call the digits 4, 5, and 6 ”mids” and digits 7, 8 and 9 ”highs”. (a) How many diﬀerent strings can he type? (b) How many of these strings have no mids? (c) How many of these strings have only one high in them? For example, the string 1111199 has two highs in it. (d) What’s the probability that a string starts with a low and ends with a high? (e) What’s the probability that a string starts with a low or ends with a high? (f) What’s the probability that a string doesn’t have at least one of the digits 1 through 9? 5. School of Probability and Statistics (SPS) at IUA University has 13 male Moroccan professors, 8 female Moroccan professors, and 12 nonMoroccan professors. A committee of 9 professors needs to be appointed for a task. (a) How many committees can be made? (b) What’s the probability 1

that the committee contains 2 Moroccan women, 3 Moroccan men, and 4 non-Moroccans? (c) What’s the probability that the committee contains exactly 4 nonMoroccans? (d) What’s the probability that the committee contains at least 4 nonMoroccans? (e) What’s the probability that the committee does not contain any Moroccan men?

Conditional Probability, Bayes’ Theorem

1. Before the distribution of certain statistical software every fourth compact disk (CD) is tested for accuracy. The testing process consists of running four independent programs and checking the results. The failure rate for the 4 testing programs are, respectively, 0.01, 0.03, 0.02 and 0.01. (a) What is the probability that a CD was tested and failed any test? (b) Given that a CD was tested, what is the probability that it failed program 2 or 3? (c) In a sample of 100, how many CDs would you expect to be rejected? (d) Given a CD was defective, what is the probability that it was tested? 2. A regional telephone company operates three relay stations at diﬀerent locations. During a one-year period, the number of malfunctions reported by each station and the causes are shown below: Station Problems with electricity supplied Computer malfunction Malfunctioning electrical equipment Caused by other human errors A 2 4 5 7 B 1 3 4 7 C 1 2 2 5

Suppose that a malfunction was reported and it was found to be caused by other human errors. What is the probability that it came from station C? 3. Police plan to enforce speed limits by using radar traps at 4 diﬀerent locations within the city limits. The radar traps at each of the locations L1 , L2 , L3 , and L4 are operated 40%, 30%, 20%, and 30% of the time, and if a person who is speeding on his way to work has probabilities 2

0.2, 0.1, 0.5 and 0.2, respectively, of passing through these locations, what is the probability that he will receive a speeding ticket? You can assume that the radar traps operate independently of each other. 4. Jar A contains 6 red balls and 6 blue balls. Jar B contains 4 red balls and 16 green balls. A six-sided die is thrown. If the die falls ”6”, a ball is chosen at random from jar A. Otherwise, a ball is chosen from Jar B. If the chosen...