I've been learning about probability lately and finding out some interesting things. I would reckon it not probable that many people will find these puzzles interesting enough to post their attempts at solutions in this thread, but I will give it a try. The second one is just complicated (hence the use of a multiple-choice format); the third is comparatively simple but potentially extremely difficult.
1. A certain family contains two children. One of them is a girl. What is the probability that the other is a girl?
2. Multiple-choice question: There is a certain disease which one person in a thousand has. There is a test to detect the presence of this disease, but it is not infallible. In 99% of cases of persons tested who have the disease, it gives a positive result (a true positive); in the other 1% of cases, it gives a negative result (a false negative). Also, in 2% of cases of persons who do NOT have the disease, it gives a positive result (a false positive). You have taken the test and gotten a positive result. How probable is it, given this information, that you have the disease? Approximately
(a) 5%
(b) 75%
(c) 98%
(d) 99%
3. You and I are going to play a game. This is how it works: I have placed a valuable gold coin underneath one of three cups. I know which cup it is under, but you don't. If you can guess which cup hides the coin, you get the coin; but the game proceeds in two steps. First, you make an initial guess as to which cup hides the coin. I will then lift one of the other two cups, to show you that the coin is not under that cup. You will then have the opportunity to switch your guess to the other remaining cup, or to stick with your original guess. For instance, suppose you guess that the coin is under cup 2, and I show you that it is not under cup 1, and give you the choice of sticking with cup 2 or switching to cup 3. Question: Do you improve your chance of winning by switching (and if so, how)?
(People who are already familiar with the third problem, which is commonly presented in different terms, are asked not to identify it by name.)
1. A certain family contains two children. One of them is a girl. What is the probability that the other is a girl?
2. Multiple-choice question: There is a certain disease which one person in a thousand has. There is a test to detect the presence of this disease, but it is not infallible. In 99% of cases of persons tested who have the disease, it gives a positive result (a true positive); in the other 1% of cases, it gives a negative result (a false negative). Also, in 2% of cases of persons who do NOT have the disease, it gives a positive result (a false positive). You have taken the test and gotten a positive result. How probable is it, given this information, that you have the disease? Approximately
(a) 5%
(b) 75%
(c) 98%
(d) 99%
3. You and I are going to play a game. This is how it works: I have placed a valuable gold coin underneath one of three cups. I know which cup it is under, but you don't. If you can guess which cup hides the coin, you get the coin; but the game proceeds in two steps. First, you make an initial guess as to which cup hides the coin. I will then lift one of the other two cups, to show you that the coin is not under that cup. You will then have the opportunity to switch your guess to the other remaining cup, or to stick with your original guess. For instance, suppose you guess that the coin is under cup 2, and I show you that it is not under cup 1, and give you the choice of sticking with cup 2 or switching to cup 3. Question: Do you improve your chance of winning by switching (and if so, how)?
(People who are already familiar with the third problem, which is commonly presented in different terms, are asked not to identify it by name.)