CSC/MTH 231 Discrete Structures II Spring, 2010 Homework 5 Name 1. A six sided die D (with sides numbered 1, 2, 3, 4, 5, 6) is thrown once. a. What is the probability that a 3 is thrown? b. What is the probability that a 7 is thrown? 2. A six sided die D (with sides numbered 1, 2, 3, 4, 5, 6) is thrown twice and each time the value thrown is recorded. a. How many trials are performed? b. What is S, the set of possible outcomes for each trial (i.e. on one single throw of one die)? c. What is S (i.e. the number of different possible outcomes) for each trial? d. If E 1 is the event that 3 the number thrown on any one trial, what is E 1 (i.e. how many different outcomes cause event E 1 to occur)? e. What is the probability on any given trial that event E 1 occurs? f. What is the probability that E 1, the complimentary event of E 1, occurs? g. If E 2 is the event that the number thrown is odd, what is the probability on any given trial that Event E 2 occurs? h. What is the probability the event E 2 occurs twice in a row (i.e. on the first trial and the second trial)? i. What is the probability on any given trial that events E 1 and E 2 both occur? j. What is the probability on any given trial that either event E 1 or event E 2 occurs? k. What is the probability that the sum of the number thrown on the first trial and the number thrown on the second trial = 6?
3. Exercise Set 6.1 (page 399) # 16 What is the probability that a 5-card poker hand contains a flush? # 20 What is the probability that a 5-card poker hand contains a royal flush (i.e. A-K-Q-J-10 of the same suit)? # 24A Find the probability of winning the lottery by selecting the correct 6 integers, where order does not matter, from the positive integers 1-30. # 28 In a super lottery a player selects 7 numbers out of the first 80 positive integers. What is the probability that a person wins the grand prize by picking 7 numbers that are among the 11 numbers selected at random by a computer? # 38 Two events E 1 and E 2 are call independent if p(e 1 E 2 ) = p(e1) * p(e 2 ). For each of the following pairs of events, which are subsets of the set of all possible outcomes when a coin is tossed 3 times, determine whether or not they are independent. a) E 1 : the first coin comes up tails; E 2 : the second coin comes up heads b) E 1 : the first coin comes up tails; E 2 : two, and not three, heads come up in a row c) E 1 : the second coin comes up tails; E 2 : two, and not three, heads come up in a row
4. Look at problem 35 on page 399. Its answers are in the back of the book. Then fill in the blanks and answer the questions below. Event E = winning on a trial (i.e. a spin) A particular casino pays you double the amount that you bet if you bet that red will come up and it does. (That is, if you bet $1 on RED and the wheel does land on a RED number, you will get $2 back, your original $1 plus another $1.) P(E) = that you win $1 & P(E ) = that you lose $1. It pays you 6 times your bet if you bet that the wheel will land on one of the numbers between 1 and 6 inclusive, and it does. (If you bet $1, you will get $6 back, your original $1 plus another $5.) P(E) = that you win $5 & P(E ) = that you lose $1. It pays you 36 times your bet if you bet that the wheel will land on a specific number (say 14) and it does. (If you bet $1, you will get $36 back, your original $1 plus another $35.) P(E) = that you win $35 & P(E ) = that you lose $1. a. Which of these 3 options gives you the greatest probability of winning at least something on any given spin? b. Which of these 3 options gives you the most money back on a spin of the wheel, relative to the probability of winning on that spin, or are they all the same? c. What is the probability that the "house" wins on any given spin? d. Over the course of a night, with hundreds of people playing roulette, how much does the house win, on average, for every dollar bet?
5. Variation on the Monty Hall "door" puzzle introduced on page 398. There are 5 doors. One has a great prize behind it and 4 have nothing behind them. You pick a door. (Let's call it door 'A'.) Then the host randomly eliminates 2 of the other 4 doors that have nothing behind them. (Let's call them doors 'D' and 'E'.) So there is now the door you selected ('A') and two other doors ('B' and 'C'). You are now given the choice of sticking with the door you originally chose or picking a different door from the remaining doors. a. What should you do? b. What is the probability you have picked the winning door if you stay with door 'A'? c. What is the probability of winning if you change your selection and pick either door 'B' or door 'C' instead? 6. Exercise set 6.1 Problem # 41 (page 400). The answers are in the back of the book, but you must show your work to indicate how you arrive at the correct answer. This is the last problem in the section. It has 3 parts (a,b,c) and begins "This problem was posed by the Chevalier de Mere ")
7. Exercise set 6.2 (pages 415-416) a. # 16 Show that if E and F are independent events, then E' and F' are also independent events. b. # 18a What is the probability that 2 people chosen at random were born on the same day of the week? c. # 26 Let E be the event that a randomly generated bit string (i.e. made up of 0s and 1s) of length 3 contains an odd number of 1s and F be the event that the string starts with 1. Are E and F independent? Show why or why not. d. # 28 Assume that the probability a child is a boy is 0.51 and that the sexes of children born into a family are independent. What is the probability that a family of 5 children has: a) exactly 3 boys b) at least 1 boy c) at least 1 girl d) all children of the same sex e. # 34 Find each of the following probabilities when n independent Bernoulli trials are carried out with a probability of success p. a) the probability of no successes b) the probability of at least one success c) the probability of at most one success d) the probability of at least 2 successes
8. Exercise set 6.3 (page 424) a. # 2 Suppose that E and F are events in a sample space and p(e) = 2/3, p(f) = 3/4, and p(f E) = 5/8. Find p(e F) b. # 4 This is the problem that begins "Suppose that Ann selects a ball ". c. # 6 This is the problem that begins "When a test for steroids is given to soccer players ".
Where possible, show your work as well as the answer. 9. Exercise set 6.4 (pages 439-441) a. # 2 What is the expected number of heads that come up when a fair coin is flipped 10 times? Mapping of random variable f(s) s f(s) tails 0 average number of heads = heads 1 b. # 4 A coin is biased so that the probability a head comes up when it is flipped is 0.6. What is the expected number of heads that come up when it is flipped 10 times? average number of heads = c. # 8 What is the expected sum of the numbers that appear when 3 fair dice are rolled? d. # 12b Suppose we roll a die until a 6 comes up. (not required) What is the probability that we roll the die n times? What is the expected number of times we roll the dice? (Hint: Use a geometric distribution.) 10. A random variable X maps outcomes in a sample space S to the following values with the probabilities shown here. -1 0 4 5 6 1/8 2/8 3/8 1/8 1/8 a. What is the mean value (i.e. average or expected value) of X? b. What is the variance of X? c. What is the standard deviation of X?
11. Chapter 6 Supplementary Exercises (page 443) # 6 A dodecahedral die has 12 faces numbered 1 12. a. What is the expected value of the number that comes up when a fair dodecahedral die is rolled. (i.e. the average value from a set of rolls) b. What is the variance of the number that comes up when a fair dodecahedral die is rolled? (i.e. the variance of the observed average value rolled from the expected average)