Assume that the box contains 9 balls:

1 red, 2 blue, and 6 yellow. As in the text, you draw one
ball, note its color, and if it is yellow replace it. If it is not yellow
you do not replace it. You then draw a second ball and note its color.
1) What is the probability that the second ball drawn is yellow?

(2) What is the probability that the second ball drawn is red?

To answer these questions, we need to determine the number of possible outcomes and the number of favorable outcomes for each scenario.

First, let's consider the probability that the second ball drawn is yellow.

Step 1: Determine the number of possible outcomes
At the beginning, the box contains 9 balls. For the first draw, any of the 9 balls could be chosen.

Step 2: Determine the number of favorable outcomes
If the first ball drawn is yellow, it is replaced. So after the first draw, there are still 9 balls in the box, out of which 6 are yellow. In this case, the probability of drawing a yellow ball on the second draw is 6/9.

If the first ball drawn is not yellow, it is not replaced. So after the first draw, there are now only 8 balls in the box, out of which 5 are yellow. In this case, the probability of drawing a yellow ball on the second draw is 5/8.

Step 3: Calculate the probability
To calculate the overall probability, we need to consider both cases and their respective probabilities.

Case 1: The first ball drawn is yellow (probability: 1/9)
The probability of drawing a yellow ball on the second draw is 6/9.

Case 2: The first ball drawn is not yellow (probability: 8/9)
The probability of drawing a yellow ball on the second draw is 5/8.

Overall probability = (Probability of Case 1) * (Probability of drawing a yellow in Case 1) + (Probability of Case 2) * (Probability of drawing a yellow in Case 2)

= (1/9) * (6/9) + (8/9) * (5/8)
= 6/81 + 40/72
= 6/81 + 5/9
= 6/81 + 45/81
= 51/81
= 17/27

So, the probability that the second ball drawn is yellow is 17/27.

Now let's move on to the probability of the second ball drawn being red.

Step 1: Determine the number of possible outcomes (same as before)
At the beginning, the box contains 9 balls. For the first draw, any of the 9 balls could be chosen.

Step 2: Determine the number of favorable outcomes
If the first ball drawn is red, it is not replaced. So after the first draw, there are still 8 balls in the box, out of which 1 is red. In this case, the probability of drawing a red ball on the second draw is 1/8.

If the first ball drawn is not red, it is not replaced. So after the first draw, there are still 8 balls in the box, out of which 1 is red. In this case, the probability of drawing a red ball on the second draw is 1/8.

Step 3: Calculate the probability
To calculate the overall probability, we need to consider both cases and their respective probabilities.

Case 1: The first ball drawn is red (probability: 1/9)
The probability of drawing a red ball on the second draw is 1/8.

Case 2: The first ball drawn is not red (probability: 8/9)
The probability of drawing a red ball on the second draw is 1/8.

Overall probability = (Probability of Case 1) * (Probability of drawing a red in Case 1) + (Probability of Case 2) * (Probability of drawing a red in Case 2)

= (1/9) * (1/8) + (8/9) * (1/8)
= 1/72 + 8/72
= 9/72
= 1/8

So, the probability that the second ball drawn is red is 1/8.