Puzzle | Maximize probability of White Ball

You have two empty jars and 100 balls, 50 white and 50 black. Your task is to distribute all the balls between the two jars in any way you choose. After the distribution, one jar will be selected at random, and then one ball will be randomly drawn from that jar.

How should you distribute the balls to maximize the probability of drawing a white ball? 

Check if you were right - full answer with solution below.

Solution :

First, assume you divide the balls equally between the two jars: 

• Each jar gets 25 white and 25 black balls.

The probability of selecting a white ball in this case is:

(1/2 × 25/50) + (1/2 × 25/50) =1/2

To maximise the probability of selecting a white ball, we need to increase the chance in at least one jar.

• Put 1 white ball in Jar 1
• Put the remaining 49 white balls and 50 black balls in Jar 2

Now calculate the probability:

• Probability of selecting Jar 1 = 1/2
• Probability of white ball from Jar 1 = 1 (since it has only one white ball)-Contribution from Jar 1 = (1/2) × 1 = 0.5
• Probability of selecting Jar 2 = 1/2
• Jar 2 has 49 white out of 99 balls - Contribution from Jar 2 = (1/2) × (49/99) ≈ 0.247.
• Total probability:

1​/2 ​×1 +1​/2 ×49/99=1​/2+49/198​≈0.747

Therefore, the probability of drawing a white ball is approximately three-fourths or 0.75.