Applied Probability and Statistics Homework

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MATH 331 – Homework 3 Instructor: Dr. Brown 1. Imagine the following game of chance. There are four dollar bills on the table. You roll a fair die repeatedly. Every time you fail to get a six, one dollar bill is removed. When you get your first six, you get to take the money that remains on the table. If the money runs out before you get a six, you?ve lost and the game is over. Let X be the amount of your award. Find the range and the probability mass function of X. 2. The statement SOME DOGS ARE BROWN has 16 letters. Choose one of the 16 letters uniformly at random. Let X denote the length of the word containing the chosen letter. Determine the range and probability mass function of X. 3. What is the probability that a randomly chosen number between 1 and 100 is divisible by 3 given that the number has at least one digit equal to 5? 4. Suppose a family has 2 children of different ages. We assume that all combinations of boys and girls are equally likely. a. Formulate precisely the sample space and probability measure that describes the genders of the two children in the order in which they are born. b. Suppose we learn that there is a girl in the family. (Precisely: we learn that there is at least one girl.) What is the probability that the other child is a boy? c. Suppose we see the parent with a girl, and the parents tell us that this is their youngest child. What is the probability that the older child we have not yet seen is a boy? 5. A bag contains 3 kinds of dice: seven 4-sided dice, three 6-sided dice, and two 12-sided dice. A die is drawn from the bag and then rolled, producing a number. For example, the 12-sided die could be chosen and rolled, producing the number 10. Assume that each die is equally likely to be drawn from the bag. a. What is the probability that the number 6 is rolled? b. What is the probability that a 6-sided die was chosen, given that the number 6 was rolled? 6. We choose one of the words in the following sentence uniformly at random and then choose one of the letters of that word (again uniformly at random): THE QUICK BROWN FOX JUMPED OVER THE GATE a. Find the probability that the chosen letter is O? b. Let X denote the length of the chosen word. Determine the probability mass function of X. 7. Insurance company X has two types of customers. An ordinary customer has an accident during the year with probability 1%. An accident prone customer has an accident during the year with probability 4%. 80% of the customers are ordinary customers. Pick a randomly chosen customer. Suppose she had an accident. What is the probability that she is an ordinary customer? 8. Let A and B be events with these properties: 0 < Pr(B) < 1 and Pr(A|B) = Pr(A|B c ) = 1 3 a. Is it possible to calculate Pr(A) from this information? Either declare that it is not possible, or find the value of Pr(A). b. Are A and B independent, not independent, or is it impossible to determine? 9. We choose a number between 1 and 100 uniformly at random, we denote it by X. For each of the following choices decide whether the two events in question are independent or not. a. A = {X is even}, B = {X is divisible by 5}. b. C = {X has two digits}, D = {X is divisible by 3}. c. E = {X is a prime}, F = {X has a digit 5}. 10. Three events A, B and C satisfy the following: A and B are independent, C is a subset of B, C is disjoint from A, Pr(A) = 1/2, Pr(B) = 1/4 and Pr(C) = 1/10. Compute Pr(A ∪ B ∪ C).
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