A bag contains 4 balls. Two balls are drawn at random, and are found to be white. What is the probability that all balls are white?

To determine the probability that all balls in the bag are white, given that two drawn balls are white, we can use Bayes' theorem. Let's break it down step by step.

Understanding the Scenario

We have a bag with 4 balls. The possible configurations of the balls can be:

• 4 white balls
• 3 white and 1 non-white ball
• 2 white and 2 non-white balls
• 1 white and 3 non-white balls
• 0 white balls

Calculating Probabilities

We need to find the probability of having all 4 balls white given that we drew 2 white balls. Let's denote:

• A: Event that all 4 balls are white.
• B: Event that 2 balls drawn are white.

Using Bayes' theorem, we want to find P(A|B), which is the probability of A given B. This can be calculated as:

P(A|B) = (P(B|A) * P(A)) / P(B)

Finding Each Probability

1. **P(B|A)**: If all balls are white, the probability of drawing 2 white balls is 1 (since all are white).

2. **P(A)**: Assuming each configuration is equally likely, the probability of having all 4 balls white is 1 out of 5, or 0.2.

3. **P(B)**: This is the total probability of drawing 2 white balls from any configuration. We calculate it as follows:

• From 4 white balls: P(B) = 1 (probability of drawing 2 white)
• From 3 white and 1 non-white: P(B) = 3/6 = 0.5
• From 2 white and 2 non-white: P(B) = 6/12 = 0.5
• From 1 white and 3 non-white: P(B) = 0 (impossible)
• From 0 white: P(B) = 0 (impossible)

Now, we calculate P(B):

P(B) = (1 * 0.2) + (0.5 * 0.2) + (0.5 * 0.2) + (0 * 0.2) + (0 * 0.2) = 0.2 + 0.1 + 0.1 = 0.4

Final Calculation

Now we can substitute these values into Bayes' theorem:

P(A|B) = (1 * 0.2) / 0.4 = 0.5

Conclusion

The probability that all balls in the bag are white, given that two drawn balls are white, is 0.5 or 50%.