Math 4653, Section 001 Elementary Probability Fall Week 3 Worksheet

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Brooke Cameron
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1 Week 3 Worksheet Ice Breaker Question: What is the first thing you bought with your own money? 1. If S n = binomial(n, p), find var(s n ). 2. Suppose a lottery ticket has probability p of being a winning ticket, independently of other tickets. A gambler buys 3 tickets, hoping that this will triple his chances of winning. (a) What is the distribution of how many of the 3 tickets are winning tickets? (b) Was the gambler s strategy a good one? 1

2 3. Charles claims he can tell the difference between green veggie straws and veggie straws of another color 75% of the time. Ruth bets that he can t and that he just guesses. To settle this, Ruth and Charles make a bet. Charles is given 10 veggie straws. He wins the bet if he correctly identifies 7 or more as green/not green. (Side note: I think the green veggie straws taste terrible and all the others taste same. I said this one time and everyone else was very skeptical because they thought all the veggies straws tasted the same. But then another grad student in the department agreed with me and he did a taste test to prove that he could taste the difference between the green veggie straws and the others.) (a) Find the probability Charles wins if he has the ability he claims. (b) Find the probability Ruth wins if Charles is guessing. 4. In men s tennis, the winner is the first to win 3 (out of 5) sets. Suppose John Isner and Kevin Anderson are playing (did anyone watch that match at Wimbledon this year?). (a) If John Isner has a 1/3 chance of winning each set, what is the probability he wins the match? 2

3 (b) What if Isner had a probability p chance of winning each set? (c) Suppose now we have players A and B playing a game. Player A has a probability p chance of winning each time. If A and B are playing best k out of 2k 1, what is the probability that A wins? 5. Each time Rick goes to out to lunch there is a 80% chance he goes to Bona, a 10% chance he goes to Punch, and a 10% chance he goes to Afro Deli. If he goes out to lunch 4 times this week, what is the probability he went to Bona twice, Punch once, and Afro Deli once? 6. If X = Poisson(λ) show that E(X(X 1)) = λ 2. (Hint: Use the same method we used to show EX = λ.) 3

4 7. In a class of 80 students, the professor calls on 1 student chosen at random each class period. There are 32 class periods in a term. (a) Write a formula for the probability a given student is called upon j times during the term? (b) Write a formula for the Poisson approximation for this probability. Using this formula, estimate the probability that a given student is called upon more than twice. 8. The probability of triplets in human births is approximately What is the probability that there will be exactly 1 set of triplets among 700 births in a large hospital? 4

5 9. Assume that the probability that there is a significant accident in a nuclear power plant during one year s time is If a country has 100 nuclear plants, estimate the probability that there is at least one such accident during a given year. 10. Which of questions 7, 8, and 9 require the general Poisson approximation result rather than just the Poisson approximation to the binomial? 11. Prove ( ) 2n = n n j=0 ( ) n 2. j (Hint: Consider an urn with n red balls and n blue balls. Show each side of the equation is the number of ways to choose n balls from the urn.) 5

6 12. A poker hand has 5 cards. What is the probability of getting four of a kind (four of one value and 1 of another) when you are dealt your hand? 13. Suppose I have $100 worth of quarters, enough to make 10 quarter rolls. Suppose I make all the quarter rolls and have 2 quarters leftover. I conclude that 2 of the rolls only have 39 quarters. If I give 2 of the quarter rolls to a friend, what is the probability the friend got exactly 1 of the rolls that is short a quarter? 14. Suppose you have a drawer with 6 white socks, and 2 black socks. If you choose 2 socks at random, what is the chance you got a matching pair? Real World Example: The court case De Martini v. Power dealt with a case of ballot irregularities in a small election. De Martini received 2,656 people votes and Power received 2954 votes. Later, 136 of the 5250 votes cast were declared invalid due to irregularities. Power wanted to overturn the election results, as his loss was only by 62 votes. However, in order to reverse the outcome of the election, as least 99 of the 136 fraudulent votes must have been cast for De Martini. Modeling this as an urn problem, we can see that ) P (m of the irregular votes were counted for De Martini) = ( 2656 )( 2594 m 136 m ( ). Summing from m = 99 to 136, we find the probability that at least 99 of the irregular votes were for De Martini is Based on this, the court decided that the election result would stand. For more information: Example 2.40 from your textbook Anita M. De Martini v. James M. Power, in the Court of Appeals of the State of New York (you can read the decision at law.justia.com) 6

The Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.)

The Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.) The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you $ that if you give me $, I ll give you $2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If