Problem Solving

I posted two different replies to that question here: https://www.beatthegmat.com/p-yellow-or- ... 94007.html

Cheers,
Brent

nahid078 wrote:A box contains 1 blue ball, 1 yellow ball, and 2 red balls.
Three balls are randomly selected (one after the other) without replacement.
What is the probability that the 2nd ball is yellow or the 3rd ball is NOT yellow?

A) 1/2
B) 3/4
C) 5/6
D) 11/12
E) 1

We are given that a box contains 1 blue ball, 1 yellow ball, and 2 red balls. We need to determine, when selecting 3 balls from the box, the probability that the second ball is yellow OR the third ball is not yellow.

Let's look at the requirements in two separate scenarios:

Scenario 1: The second ball is yellow

The probability of not choosing the yellow ball in the first draw is 3/4, and the probability of choosing the yellow ball from the remaining 3 balls in the second draw is 1/3 (we do not care which ball is selected in the third draw in this scenario). Therefore, the probability of choosing the yellow ball in the second draw is:

3/4 x 1/3 = 1/4

Scenario 2: The third ball is not yellow. There are several ways this can happen, as follows:

a. The first ball is yellow. If this happens, we know that the third ball cannot be yellow because there is only one yellow ball in the box.

Since there are four balls, the probability of choosing the yellow ball in the first draw is 1/4 (we do not care which balls are selected in the second or third draws in this scenario).

b. The second ball is yellow. If this happens, we know that the third ball cannot be yellow because there is only one yellow ball in the box. We already considered this outcome in Scenario 1, so we WILL NOT include it again here.

c. The yellow ball is not selected at all. This outcome also satisfies the requirement that the third ball is not yellow.

The probability of choosing a non-yellow ball in the first, second, and third draws is 3/4, 2/3, and 1/2, respectively. So the probability of not choosing a yellow ball in three draws is:

3/4 x 2/3 x 1/2 = 1/4

Since any of these outcomes would satisfy the given condition and they are mutually exclusive, we can sum the individual probabilities of each outcome:

1/4 + 1/4 + 1/4 = 3/4

Answer: B

nahid078 wrote:A box contains 1 blue ball, 1 yellow ball, and 2 red balls.
Three balls are randomly selected (one after the other) without replacement.
What is the probability that the 2nd ball is yellow or the 3rd ball is NOT yellow?

A) 1/2
B) 3/4
C) 5/6
D) 11/12
E) 1

Another solution is to list all of the possible outcomes for the 3 selections:
- RRY
- RRB
- RBR
- RYR
- RYB
- RBY
- BYR
- YBR
- BRY
- YRB
- BRR
- YRR

Of the 12 possible outcomes, 9 meet one or both of requirements.
So, P(2nd ball is yellow or the 3rd ball is NOT yellow) = 9/12 = 3/4 = B