Wanna Bet? Interesting Puzzles from Betting and Probability

Transcription

1 Wanna Bet? Interesting Puzzles from Betting and Probability Saturday and Sunday, November 20-21, 2010 MIT Room W Sat 3pm-5pm MIT Room Sun 10am-12pm Stephen M. Hou Splash 2010 Educational Studies Program Massachusetts Institute of Technology Assumptions 1. A standard deck of cards has 26 black cards and 26 red cards (spades and clubs are black, hearts and diamonds are red). 2. A standard die has six faces numbered 1 through 6. When rolled, each face is equally likely to appear. 3. Money is infinitely divisible (i.e. you are not restricted to multiples of cents). 4. All bets are fair. Suppose you bet me $x on outcome A. If outcome A occurs, I pay you $x. If not, you pay me $x. 5. You cannot bet more than you have Nov 2010 Splash Wanna Bet? - S.M. Hou 2

2 Prerequisites To be fully prepared for this class, you should be able to solve the following two problems. 1. What is the probability that exactly two heads will come up if I flip a fair coin three times? 2. What is the probability that a sum of 8 appears if I roll a pair of standard dice? Nov 2010 Splash Wanna Bet? - S.M. Hou 3 Cards on Foreheads: Extreme Cases Several students are sitting in a circle. Their teacher places a random playing card on each of their foreheads. The students cannot see their own card but they can see everyone else s. They also cannot communicate with one another once the cards are up. The students must simultaneously guess the color (black or red) of their own card. If either everyone guesses correctly or everyone guesses incorrectly, the teacher will give them all As. Otherwise, they all get Fs. The students have an opportunity to discuss strategy with one another before the teacher puts up the cards. Can the students guarantee that they will all get As? How? What if, instead of guessing the color, the students are to guess the suit of the card? Nov 2010 Splash Wanna Bet? - S.M. Hou 4

3 Cards on Foreheads: Hedged Case In the previous problem, we wanted to obtain the extreme results (i.e. all right or all wrong). Now, let s try the middle situation Several students are sitting in a circle. Their teacher places a random playing card (with replacement) on each of their foreheads. The students cannot see their own card but they can see everyone else s. They also cannot communicate with one another once the cards are up. The students must simultaneously guess the color (black or red) of their own card. If exactly half of them guess correctly, teacher will give them all As. Otherwise, they all get Fs. (Assume there are an even number of students.) The students have an opportunity to discuss strategy with one another before the teacher puts up the cards. Can the students guarantee that they will all get As? How? Nov 2010 Splash Wanna Bet? - S.M. Hou 5 World Series Betting World Series Baseball: Two teams play against each other until one wins four games, or equivalently, best out of seven (no ties). Your uncle gives you $2,000 to bet on his behalf in favor of one of the two teams. Thus, if his team wins the series, you need to return him $4,000, and if his team loses, you don t need to return him anything. Unfortunately, you cannot find anyone to bet on the entire series against you. You can, however, always find someone to bet against you any amount of money, up to your current balance, on any individual game, after which your balance is adjusted accordingly. There is a series of bets you can make to guarantee that you can meet your uncle s wishes. How much must you bet on the first game in order to do so? Try it with fewer games instead (e.g. best out of three ). There is only one correct answer. Don t try to beat the game (i.e. end up with more than $4,000 if your uncle s team wins). Just do the minimum Nov 2010 Splash Wanna Bet? - S.M. Hou 6

4 An Extra Coin Player A has one more coin than player B (B has n coins and A has n+1 coins). Both players throw all of their coins simultaneously and observe the number that come up heads. All the coins are fair. What is the probability that A obtains more heads than B does? There is an easy way and a hard way to solve this problem. The hard, but straightforward, approach is to look at all the possibilities and calculate. The easy, but clever, approach is to use symmetry. For reference, here are some answers for similar questions: If A and B each have one coin, the probabilities are: 5/16 that A gets more heads than B 5/16 that B gets more heads than A 3/8 that A and B get the same number of heads If A and B each have 10 coins, the probabilities are: 41.2% that A gets more heads than B 41.2% that B gets more heads than A 17.6% that A and B get the same number of heads Try it for some small n. Consider the case where A has several more coins than B. Then see what happens when A s number of coins exceeds B s by just one Nov 2010 Splash Wanna Bet? - S.M. Hou 7 The Highest Die I play the following game with you. You simultaneously roll three standard dice. If the highest numbered face rolled is x, I pay you $x. What are your expected (average) earnings? How much would you pay for the privilege of playing this game? What is the probability that the highest numbered face rolled is a 6? How about a 5? 4? 3? 2? 1? The answer is not 1/6. Which number should have the highest probability? Lowest probability? Use a weighted average Nov 2010 Splash Wanna Bet? - S.M. Hou 8

5 Two Dice A standard die is labeled 1, 2, 3, 4, 5, 6 (one integer per face). When you roll two standard dice, it is easy to compute the probability of the various sums. For example, the probability of rolling a 2 (or a 12) is 1/36, a 3 (or 11) is 2/36=1/18, a 4 (or 10) is 3/36=1/12, a 5 (or 9) is 4/36 = 1/9, a 6 (or 8) is 5/36, and finally a 7 is 6/36 = 1/6. Find a pair of six-sided non-standard dice (possibly different from each other) with positive integer labels that produce the same possible sums as a pair of standard dice with the same probabilities. These non-standard dice may have the same integer on several faces (i.e. the faces need not be distinct). Try guess-and-check. Think about the number of 1s each die must have. Then think about the possible maximum values on each die. Or, consider the function f(x) = (1x 1 + 1x 2 + 1x 3 + 1x 4 + 1x 5 + 1x 6 ) 2 = 1x 2 + 2x 3 + 3x 4 + 4x 5 +5x 6 + 6x 7 + 5x 8 +4x 9 + 3x x x Nov 2010 Splash Wanna Bet? - S.M. Hou 9 The Duel Ben and Rick decide to settle their differences with a pistol duel. They take turns shooting each other until someone is hit. The one who fired the hitting shot is the winner. Ben s and Rick s shots hit their targets 30% and 25% of the time, respectively. Each shot is completely independent of the others. Ben, the better shot, suggests they flip a coin to determine who shoots first. Rick insists that he should shoot first because he s the weaker shot. Ben counters that Rick going first would overcompensate. If Rick shoots first, what is the probability that he is the ultimate winner? To Think About After Class: If they flip a coin to determine who shoots first, what is the probability that Rick is the ultimate winner? Nov 2010 Splash Wanna Bet? - S.M. Hou 10

6 City Census A city is interested in determining the average number of children per household in that city. Assume all households in the city have at least one child, and all children attend school in the city. The city employs two methods to do this. First, the city sends a census taker to go from house to house to observe how many children are in each household. Assume all households are accounted for and that the observations are accurate. This average number is 2. Second, the city asks all schools to ask their students to report the number of children in their household (including themselves). Assume all students are accounted for and tell the truth. This average number is 2.5. How is this possible? Why is there a discrepancy in the results of the two surveys? Try this with an example set of households. Think carefully about how the average is taken Nov 2010 Splash Wanna Bet? - S.M. Hou 11 Two Heads in Row You flip a coin until you get two heads in a row. Thus, possible sequences include: HH HTHH HTTHTHH THTTHTTHH Note that the number of flips it takes varies from two to infinity. What is the expected (average) number of flips it takes to get two heads in a row? There is an easy way and a hard way to solve this problem. The hard, but straightforward, approach is to look at all the possibilities and calculate. What is the easy, but clever, approach? Nov 2010 Splash Wanna Bet? - S.M. Hou 12

7 Feedback By the end of this weekend, these slides will be available online: Splash course catalog (for those officially registered) I will send an to all of you asking for your feedback. What did you think of this class? What can be improved for next time? Contact me: Stephen Hou stephenhou@alum.mit.edu Following this slide are some additional problems you can try at home. Enjoy! Nov 2010 Splash Wanna Bet? - S.M. Hou 13 Getting Asked to the Prom You are a shy boy waiting for girls to ask you to the senior prom, which is just four days away. Each day, starting today and ending the day before the prom or until you get a date, exactly one random girl will ask you to be her date. By the end of the day, you must either accept or reject her. If you accept a date, you must go with her. You cannot hold a girl as a back up in case someone better asks you later. That s mean! Assume that the date quality of each girl in your school is a real number randomly and uniformly distributed from 0 to 1. You know the quality of each girl, but you don t know who will ask you until she actually asks. What strategy for accepting and rejecting girls maximizes the expected (average) quality of your date? What is this value? If you wait until the last day, you have no choice but to accept the girl who asks you that day. What is her expected quality? Your strategy should be of the form: If it is n days before the prom, I will accept a girl whose quality is above some threshold x. Intuitively, you can afford to be picky in the beginning, but you get increasingly desperate as the prom approaches, so your threshold gradually lowers as you wait longer and longer. For the curious: You need at least 16 days for your expected date quality to exceed 0.9, 35 days to exceed 0.95, and 192 days to exceed Nov 2010 Splash Wanna Bet? - S.M. Hou 14

8 How Much Would You Pay? Linear Case I invite you to participate in the following gambling game. I keep flipping a fair coin until I get a head. If it took me n flips to get that head, I pay you n dollars. Thus, if it took me a single flip (probability ½), then I pay you $1. If it took me two flips (tails, then heads; probability ¼), then I pay you $2. How much would you pay for the privilege of playing this game? Nov 2010 Splash Wanna Bet? - S.M. Hou 15 How Much Would You Pay? Exponential Case This time, I invite you to participate in the following slightly different gambling game. I keep flipping a fair coin until I get a head. If it took me n flips to get that head, I pay you 2 n dollars. Thus, if it took me a single flip (probability ½), then I pay you $2. If it took me two flips (tails, then heads; probability ¼), then I pay you $4. How much would you pay for the privilege of playing this game? Nov 2010 Splash Wanna Bet? - S.M. Hou 16

9 War of Attrition You run a military boot camp that is so arduous that at the end of each day, 5% of the remaining participants drop out. This means that 5% lasted only one day, 95%*5% lasted only two days, etc. What is the expected (average) number of days participants last in your boot camp? Note: If you changed the 5% to 50%, the problem is identical to average number of flips of a coin needed to get a head, or the average number of children needed to get a girl Nov 2010 Splash Wanna Bet? - S.M. Hou 17 Birthday Line At a movie theatre, the manager announces that they will give a free ticket to the first person in line whose birthday is the same as someone who has already bought a ticket. You are able to get in line with any number of people ahead of you, which you must do before the cashier starts to ask each patron's birthday one by one as they buy their tickets. Assume that no one knows anyone else's birthday and that birthdays are distributed randomly and uniformly throughout the year. (It turns out that it doesn't matter whether February 29th exists.) What position in line gives you the greatest chance of being the first duplicate birthday and thus the winner of the free ticket? If you are given even greater freedom and can change your position in line as non-winners are revealed one by one, should your strategy change? How? Hint: Start with smaller numbers of possible birthdays, and then gradually increase it to Nov 2010 Splash Wanna Bet? - S.M. Hou 18

10 Comparing Numbers You and I play the following game. You use a random number generator to obtain two real numbers independently and uniformly chosen from zero to one. You choose one of the numbers to show me. I then guess whether this number is smaller or larger than the concealed random number. What strategy should you use to decide which number to show me so that I cannot guarantee having a greater than 50% probability of guessing correctly? Your strategy should be such that EVEN IF I knew that strategy, I still wouldn't be able to guarantee having a greater than 50% probability of guessing correctly. A reasonable guessing rule I could use is as follows: If the number shown me is greater than 0.5, guess that it is larger, else guess that it is smaller. Let's think about how my rule would fare against possible strategies you could use. If you always choose to show me the larger of the two numbers (and assuming I don't know you are doing this, of course), then by employing the above rule, my success rate would be 75%. Can you see why? Similarly, if you always choose to show me the smaller of the two numbers, my success rate would also be 75%. If you are lazy and simply pick one of the two numbers at random with a fair coin flip, then my success rate would still be 75%. Can you see why? From your point of view, this strategy better than the above two because even if I knew your strategy, my success rate would remain at 75%, instead of shooting up to 100% in the case of the other two Nov 2010 Splash Wanna Bet? - S.M. Hou 19 Counting Cards You start with $1. I shuffle a standard deck of cards. I flip the cards up one at a time. Prior to each flip, you can bet any fraction of your current worth on the color (red or black) of the next card. After each flip, you and I transfer money accordingly. What is the highest amount you can guarantee you ll end up with? How do you achieve this? You can guarantee $2 by only betting on the last card. The answer is more than $2. If you are lucky (you always bet everything and you are always right), you can end up with $2^52, or 4.5 quadrillion dollars. The answer is around $9.08, which is (2^52) divided by (52 choose 26). Now that you know the answer, think about how to achieve it Nov 2010 Splash Wanna Bet? - S.M. Hou 20