Probability Exercice
MTH3301 Fall 2012 Practice problems
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
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