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 the balls in the bag are white, given that two drawn balls are white, we can use a logical approach based on conditional probability. Let’s break this down step by step.

Understanding the Problem

We have a bag containing a total of 4 balls. The problem states that two balls drawn at random are white. We need to find the probability that all the balls in the bag are white.

Setting Up the Scenario

Let’s denote the number of white balls in the bag as W and the number of non-white balls as N. Since the total number of balls is 4, we can express this as:

• W + N = 4

Now, we need to consider the possible combinations of white and non-white balls that would allow us to draw two white balls.

Possible Combinations of Balls

Here are the scenarios for the number of white balls:

• If W = 4 (all balls are white): This is a valid scenario.
• If W = 3 (three white, one non-white): This is also valid.
• If W = 2 (two white, two non-white): This is valid as well.
• If W = 1 (one white, three non-white): This is not valid because we cannot draw two white balls.
• If W = 0 (no white balls): This is also not valid for the same reason.

Calculating the Probabilities

Now, we can focus on the valid scenarios:

• W = 4: The probability of drawing two white balls is 100%.
• W = 3: The probability of drawing two white balls is calculated as:
• P(drawing 2 white from 3) = (3/4) * (2/3) = 1/2.
• W = 2: The probability of drawing two white balls is:
• P(drawing 2 white from 2) = (2/4) * (1/3) = 1/6.

Using Bayes' Theorem

To find the probability that all balls are white given that we drew two white balls, we can apply Bayes' theorem:

• P(All white | Drew 2 white) = P(Drew 2 white | All white) * P(All white) / P(Drew 2 white)

Let’s denote:

• P(All white) = 1/4 (since there are 4 equally likely combinations of W)
• P(Drew 2 white | All white) = 1 (since if all are white, we will always draw white)
• P(Drew 2 white) = P(Drew 2 white | W=4) * P(W=4) + P(Drew 2 white | W=3) * P(W=3) + P(Drew 2 white | W=2) * P(W=2)

Calculating P(Drew 2 white)

Substituting the values:

• P(Drew 2 white) = 1 * (1/4) + (1/2) * (1/4) + (1/6) * (1/4)
• P(Drew 2 white) = 1/4 + 1/8 + 1/24

Finding a common denominator (24):

• P(Drew 2 white) = 6/24 + 3/24 + 1/24 = 10/24 = 5/12.

Final Calculation

Now we can substitute back into Bayes' theorem:

• P(All white | Drew 2 white) = 1 * (1/4) / (5/12) = (1/4) * (12/5) = 3/5.

Conclusion

The probability that all the balls in the bag are white, given that two drawn balls are white, is 3/5. This means there is a 60% chance that all the balls are indeed white based on the information provided.