Boxes and envelopes Please answer all questions in complete sentences. Consider the following set-up. Mr. Jones has two children. For these questions, assume that a child must be either a girl or a boy, that girls and boys exist in equal numbers in the population, and that the gender of a child is independent of the gender of the child s sibling. 1. If the older child is a girl. What is the probability that both children are girls? 2. If at least one of the children is a boy, what is the probability that both children are boys? 1
3. Do your answers to the last two problems make sense? Explain. How is this problem similar to some previous problems we have looked at? 2
Next we have the following set-up. There are three boxes: a box containing two gold coins, a box containing two silver coins, a box containing one gold coin and a silver coin. 4. After choosing a box at random and withdrawing one coin at random, if that happens to be a gold coin, what is the probability that the next coin drawn from the same box is also a gold coin? 5. In which ways is this problem similar to the Monty Hall problem? In which ways is it different? 3
Three prisoners, A, B and C, are in separate cells and sentenced to death. The governor has selected one of them at random to be pardoned. The warden knows which one is pardoned, but is not allowed to tell. Prisoner A begs the warden to let him know the identity of one of the others who are going to be executed. If B is to be pardoned, give me C s name. If C is to be pardoned, give me B s name. And if I m to be pardoned, flip a coin to decide whether to name B or C. 6. The warden tells A that B is to be executed. Prisoner A is pleased because he believes that his probability of surviving has gone up from 1/3 to 1/2, as it is now between him and C. Prisoner A secretly tells C the news, who is also pleased, because he reasons that A still has a chance of 1/3 to be the pardoned one, but his chance has gone up to 2/3. What is the correct answer? Explain. (Hint: Compare this problem to the Monty Hall problem.) 4
7. Explain how the previous problem is similar to the Monty Hall problem. Is it exactly the same, or are there differences? 5
You are given two indistinguishable envelopes, each containing a check, one contains twice as much as the other. You may pick one envelope and keep the money it contains. Having chosen an envelope at will, but before inspecting it, you are given the chance to switch envelopes. 8. Should you switch? What is the probability of getting the larger amount of money if you switch? What is the probability of getting the larger amount of money if you stay with your first choice? 9. Suppose now that you are allowed to choose an envelop, open it, and inspect the amount of money inside before making your choice. Design a (random) strategy that makes it more likely you will be able to get the larger sum of money. (Hint: Try introducing another source of randomness.) 6
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake: If the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends. Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before. 10. During the interview Beauty is asked: What is your credence now for the proposition that the coin landed heads? How might she answer? 7
11. People have made several strange arguments for this problem. The thirder position argues that the probability of heads is 1/3. Adam Elga argued for this position originally as follows: Suppose Sleeping Beauty is told and she comes to fully believe that the coin landed tails. Her credence that it is Monday should equal her credence that it is Tuesday since being in one situation would be subjectively indistinguishable from the other. Consider now that Sleeping Beauty is told upon awakening and comes to fully believe that it is Monday. Since the coin is fair, either outcome is equally likely. Thus, since there are three total outcomes and all are equally likely, the probability of each is one-third. David Lewis responded to Elga s paper with the position that Sleeping Beauty s credence that the coin landed heads should be 1/2. Sleeping Beauty receives no new non-self-locating information throughout the experiment because she is told the details of the experiment. Others have argued that this is not true because she does obtain new information. She knows that she is now in it, but does not know whether it is Monday or Tuesday. Discuss each of the arguments above in your group. Do you agree with any of these arguments? Why or why not? Choose one of the arguments above and make a case for why it should be false. 8
Two men are each given a necktie by their respective wives as a Christmas present. Over drinks they start arguing over who has the cheaper necktie. They agree to have a wager over it. They will consult their wives and find out which necktie is more expensive. The terms of the bet are that the man with the more expensive necktie has to give it to the other as the prize. The first man reasons as follows: winning and losing are equally likely. If I lose, then I lose the value of my necktie. But if I win, then I win more than the value of my necktie. Therefore, the wager is to my advantage. The second man can consider the wager in exactly the same way; thus, paradoxically, it seems both men have the advantage in the bet. This is obviously not possible (assuming both prefer the more expensive necktie). 12. How can this apparent paradox be resolved? In other words what is wrong with the mens reasoning? 9
13. In what ways is this problem similar to other problems on the worksheet? In what ways does it differ? 10