Alice, Bob, and Charlie are all taking Spanish this semester. The instructor has

Question

Alice, Bob, and Charlie are all taking Spanish this semester. The instructor has decided that she will give A to exactly two of the three students, chosen uniformly at random, but has not released their names. Naturally, Alice figures that she will get an A with probability 2/3.

The TF offers to tell Alice the name of one of the other students who will receive an A. If the TF has a choice of naming both Bob and Charlie (because both of them will get an A), he names one of the two with equal probability.

Alice knows that the TF will tell the truth, but she declines his offer. Her reasoning is that knowing what the TF says will reduce her chances, so she is better off not knowing. For example, if the TF says that Charlie will get an A, then her probability of getting an A will decrease to 1/2 because she will then know that either she or Bob will also get an A, and these two events are equally likely.

Alice has made a typical mistake when reasoning about conditional probability. Using the tree diagram method, explain Alice’s mistake. Find the correct probability that Alice gets an A given that the TF says that Charlie gets an A.

Explanation / Answer

P( Alice gets A/ Charlie Gets A) = P( Alice and Charlie both get A)/ P( Charlie Gets A) = 2/3*2/3 /[2/3] = 2/3

Hence, conditional probability still remains 2/3 and not 1/2.

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