Maximize probability of White Ball | Puzzles for Interview - Interview Preparation PDF Download

Introduction

Imagine you have two empty bowls in a room and you have 50 white balls and 50 black balls. After you put the balls into the bowls, a random ball will be picked from a random bowl. Your goal is to distribute the balls in such a way that the chance of picking a white ball is maximized.

Equal Distribution

At first, let's assume that you divided the balls into jars equally so that each jar contains 50 balls. In this case, the probability of selecting a white ball will be calculated as follows: probability of selecting the first jar multiplied by the probability of a white ball in the first jar, plus the probability of selecting the second jar multiplied by the probability of a white ball in the second jar.

Equal Distribution Probability

This can be written as (1/2)*(25/50) + (1/2)*(25/50) = 0.5. However, since the objective is to maximize the probability of selecting a white ball, we need to increase the probability of a white ball in the first jar while keeping the probability in the second jar the same, which is equal to 1.

Maximizing Probability

To maximize the probability of selecting a white ball, add 49 white balls with 50 black balls in the first jar and only one white ball in the second jar. In this case, the probability will be (1/2)*(49/99) + (1/2)*(1/1) = 0.747. Therefore, the probability of getting a white ball becomes 1/2*1 + 1/2*49/99, which is approximately 3/4.

Conclusion

In conclusion, to maximize the chance of selecting a white ball, you need to distribute the white balls unequally in the two jars to increase the probability of selecting a white ball from the first jar. This is a great example of how probability can be manipulated by changing the distribution of items in a random selection process.