Conditional Probability and the Bertrand's Box Paradox

Conditional probability is a fundamental concept in probability theory, often illustrated through paradoxes such as Bertrand's Box Paradox. This article provides a clear explanation of the problem and the application of conditional probability to solve it.

Understanding the Problem

Imagine there are three boxes:

Box A: Contains two black balls (BB) Box B: Contains two white balls (WW) Box C: Contains one black and one white ball (BW)

You randomly select a box and then randomly draw a ball. If the ball you draw is white, what is the probability that the other ball in the same box is also white?

Solving the Problem Step-by-Step

Let's break down the solution into several steps, using conditional probability.

Step 1: Identify the Possible Scenarios

If you draw a white ball, it can only come from:

Box B: WW Box C: BW

Box A, which has two black balls, is not a possibility.

Step 2: Calculate the Probability of Selecting a White Ball from Each Box

Using the probabilities of each box being selected:

Each box has a 1/3 chance of being selected.

Box B (WW): Probability of drawing a white ball 1 (you always get a white ball). Box C (BW): Probability of drawing a white ball 0.5 (you can draw either a black or a white ball). Box A (BB): Probability of drawing a white ball 0 (impossible).

Step 3: Determine the Overall Probabilities

Let's calculate the overall probability of drawing a white ball:

Probability (PWhite|Box B) of drawing a white ball from Box B 1 * (1/3) 1/3 Probability (PWhite|Box C) of drawing a white ball from Box C 0.5 * (1/3) 1/6

Summing these probabilities gives the total probability of drawing a white ball:

PWhite PWhite|Box B PWhite|Box C 1/3 1/6 2/6 1/6 3/6 1/2

Step 4: Apply Bayes' Theorem to Find the Desired Probability

We need to find the probability that the other ball in the same box is also white, given that we drew a white ball:

PWhite|White (PWhite|Box B * PBox B) / PWhite

Substituting the known values:

PWhite|White (1 * (1/3)) / (1/2) (1/3) / (1/2) (2/3)

Conclusion

Therefore, the probability that the other ball in the same box is also white, given that you selected a white ball, is:

2/3

This problem highlights the importance of clearly defining the conditions under which information is obtained, as it can significantly affect the solution. The Bertrand's Box Paradox is a prime example of how conditional probability can lead to different interpretations and results depending on the context.

Additional Insights

Some variations of this problem, such as the Monty Hall Problem and the "Two Children" problem, offer similar challenges in determining the correct approach under different conditions. These problems emphasize the need for careful consideration of the information given and the implications of how that information is derived.

The ambiguity in Bertrand's Box Paradox underscores the necessity of clear communication in probability and statistics problems, where a small change in the phrasing can lead to a different solution.