N people in a line (N>=2) write down an Integer I (I <=100) one by one. You win if you are the first person whose number is the number of someone who writes before you. Suppose you have the option of getting position and no one knows the number of anyone else, then position in line gives you the best chance of being the winner?
Best position in line
- Thread starter tuanl
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Let's find your probability of winning if you're in position (k) on line. For you to win, the people in positions (1,...,k-1) should all have distinct numbers, and you should have one of those numbers. The probability of this happening is (P_k=\frac{99}{100}\cdot\frac{98}{100}\cdots\frac{102-k}{100}\cdot\frac{k-1}{100}). The ratio (P_{k+1}/P_k) is (\frac{(101-k)k}{100(k-1)}) which is (>1) for (k\leq 10). This means (P_1<P_2<\cdots P_{10} < P_{11} > P_{12} > \cdots), so the optimal position is 11-th in line.
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This is the birthday problem with different numbers.
