Question: Marilyn vos Savant was asked the following question for her column in Parade: You're at a party with 199 other guests when robbers break in and announce that they are going to rob one of you. They put 199 blank pieces of paper in a hat, plus one marked "you lose." Each guest must draw, and the person who draws "you lose" will get robbed. The robbers o↵er you the option of drawing first, last, or at any time in between. When would you take your turn? The draws are made without replacement, and for (a) are uniformly random.
(a) Determine whether it is optimal to draw first, last, or somewhere in between (or whether it does not matter), to maximize the probability of not being robbed. Give a clear, concise, and compelling explanation.
(b) More generally, suppose that there is one "you lose" piece of paper, with "weight" v, and there are n blank pieces of paper, each with "weight" w. At each stage, draws are made with probability proportional to weight, i.e., the probability of drawing a particular piece of paper is its weight divided by the sum of the weights of all the remaining pieces of paper. Determine whether it is better to draw first or second (or whether it does not matter); here v > 0,w > 0, and n ≥ 1 are known constants.