Drawing Balls from an Urn: Finding the Probability of Exactly Two White Balls in the First Four Draws

A ball is drawn from an urn containing three white and three black balls. After the ball is drawn, it is then replaced and another ball is drawn. This goes on indefinitely. What is the probability that of the first four balls drawn, exactly two are white?

Probability problems involving urns, balls, and replacement are quite common, and they can be quite challenging. In this blog post, we will explore the probability of drawing exactly two white balls in the first four draws from an urn containing three white and three black balls, with replacement after each draw.

To find the probability of drawing exactly two white balls in the first four draws, we need to consider all the possible outcomes and count the ones that meet our condition. Let's start by finding the total number of outcomes for the first four draws. Since there are two possible outcomes (white or black) for each draw, the total number of outcomes for the first four draws is 2^4 = 16.

Next, we need to count the number of outcomes where exactly two white balls are drawn in the first four draws. There are several cases to consider:

Case 1: WWWB, WWBW, WBWW, BWWW

In each of these cases, we have exactly two white balls in the first four draws, and they can appear in any of the first four positions. The probability of each of these cases is (3/6)^2 * (3/6)^2 * (3/6)^0 * (3/6)^1 = 3/16, and there are four such cases.

Case 2: WWBB, WBWB, BWBW, BBWW

In each of these cases, we have exactly two white balls and two black balls in the first four draws, and they can appear in any of the first four positions. The probability of each of these cases is (3/6)^2 * (3/6)^0 * (3/6)^1 * (3/6)^1 = 3/16, and there are four such cases.

Adding up the number of outcomes for each case, we get 4 + 4 = 8 outcomes that meet our condition.

Therefore, the probability of drawing exactly two white balls in the first four draws from the urn is 8/16 = 1/2, or 50%.

We calculated the probability of drawing exactly two white balls in the first four draws from an urn containing three white and three black balls, with replacement after each draw. We found that the probability is 50%, which means that it is equally likely that exactly two white balls will be drawn in the first four draws. Probability problems like this one are important because they can help us make predictions and decisions based on the likelihood of different outcomes. The concept of probability is a powerful tool that is used in many fields, including statistics, finance, and engineering.