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14. Symmetric events: Toss a fair coin $N$ times. Let $X$ be the largest between the following two numbers: the number of H's minus the number of T's or the number of T's minus the number of H's. Compute the probability mass distribution and the expected value of $X$. 13. Fair pair: Toss a pair of fair coins. What's the probability of seeing a pair of Heads once in four trials? 12. Coupon collection: On a given bubble-gum wrapper one can find one of ten famous baseball players. What is the expected number of bubble-gums you have to unwrap in order to collect all ten players? 11. Simpson's Paradox: Do the second part of the Project at the end of Chapter 6, p. 329. 10. Monty Hall randomized: Same game-show as in problem 6. below. Undecided between the strategy of always staying put or always switching, you settle on a mixed strategy: when asked if you want to switch you toss a fair coin and switch if it comes out Heads. Suppose you win the car, what is the probability that you switched? 9. Laplace's urn: An urn contains two balls that could be either white or black. Let D be the event that the balls have different colors, B that they're both black, and W that they're both white. Initially you make the assumption that P(D)=1/2, P(B)=1/4 and P(W)=1/4. Then you pick a ball at random, replace it, mix and pick again. Suppose that doing that you draw two white balls. Use this information to update your initial guess on P(D), P(B) and P(W). Do the same with a different initial guess where D,B, and W are equiprobable. 8. Polya's urn: An urn contains 5 white balls and 5 red balls. Pick a ball, note its color, then put it back in the urn and add another ball of the same color in the urn. Mix well and repeat. What is the probability that you pick a red ball on the second draw? 7. Does the king have a sister? The king comes from a family of two children. What is the probability that the other child is his sister? 6. Monty Hall: In this game-show there are three doors, one of which hides a brand new car. The participant picks a door but doesn't open it. Then the game-show host looks behind the other two and opens one of them that doesn't hide the car. The participant now has two options: either stick with the initial choice or switch to the other uncovered door. Question: is the strategy of always "switching" better than the strategy of always "staying put"? One response that is often heard is that it doesn't matter because one is ultimately faced with two doors so the chances are fifty-fifty. In reality, neither switching nor staying put has fifty-fifty odds. Consider: The strategy of "always staying" with the first choice has the same odds as the initial odds, namely a one-in-three chance of getting the car. On the other hand, "always switching" gives a two-in-three chances of winning the prize because the only way to lose by switching is if one had picked the winning door to begin with and that has a 1/3 probability. What matters here is repetition. You have to imagine the game being played over and over and a player sticking to one strategy over many games. Here are two other examples that have the same structure. A strange lottery: Suppose that you are singled out as Player 1 in a giant national game that will proceed in two stages. The lottery board will give out 26^5 different tickets, roughly 12 million, each with a string of 5 letters of the alphabet (with replacement): AAAAA, AAAAB, etc...one ticket per person. Then, in secrecy, the governor of the state will spin a wheel (with all the letters along its perimeter) five times and will determine the winning ticket. If you happen to hold the winning ticket, the board will randomly choose another ticket-holder as the second participant. If you don't have the winning ticket, then the holder of the winning ticket is chosen as the second contestant. Finally, you are paired up with this second player and you are asked if you want to switch or stay put. What would you do? Are the odds fifty-fifty? Extremely rare diseases: Suppose you have a 1 in 50,000 chances of having a certain genetic disease called knee-and-chin disease. And suppose that in the town of Manhattan KS (that does have 50,000 people) there is exactly one person with this disease. A doctor picks someone at random for Player 1. If Player 1 happens to have the disease the doctors picks the second player from the remaining 49,999 people. If Player 1 does not have the disease then Player 2 is the one person with the disease. Now Player 1 and 2 are put in the same room. Does Player 1 now have a fifty-fifty chance of having the genetic disease?
5. At a game show a team consisting of 3 people (Tom, Jane and their son Jordan) is faced with 3 doors. Behind one of the doors is a brand new minivan, behind a second door there are the keys to that van, and behind the remaining door there is a 5-gallon gas canister. The game-show host assigns to Tom the task of finding the van, to Jane the task of finding the keys, and to Jordan the task of finding the gas. Then while the rest of the family is in a sound-proof and windowless booth, each participant takes turns coming up to the doors and can open 2. If everyone completes its own assigned task, the family can drive away with the new van. Design a strategy that the family can follow so as to have 2 out of 3 chances of winning the van.
4. I write C(n,k) for "n choose k".
3. I have two 10 dollar bills. Here's my offer: I will give them to you and make a statement. If the statement is false you give me one of the bills back and keep the other. If the statement is true you get to keep both. Should you accept?
2. There's an island whose inhabitants are either honest and make only true statements or dishonest and make only false statements. An explorer in search for gold lands on this island, meets a native and asks him if there's gold on the island. The native responds "If I am honest, then there is gold on the island". What should the explorer conclude? (Assuming the explorer knows the nature of the islanders)
1. Redo problem 46 of 5.4 but with undistinguishable bottles. |