IBM Ponder This #110 · June 2007

You are playing a game in which you can ask for numbers. Each such number is a random value uniformly (and independently) distributed between 0 and 1. After you receive each number you can decide whether to request an additional number or to stop. When you stop your score is the sum of all the numbers you have received. Let 0<x<1. We ask two questions.

1. Suppose you are trying to achieve a score between x and 1. What are your chances of success?
2. Suppose you are trying to achieve a score between n+x and n+1. What are your chances of success in the limit as n --> infinity?

In both cases assume you are following the obvious best strategy. The answers are simple functions of x. Both questions must be answered correctly for credit.

Solution

Best opened after a real attempt

To be added.