A jar contains 8 red marbles and \(y\) white marbles.

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Manhattan Prep

A jar contains 8 red marbles and \(y\) white marbles. If Joan takes 2 random marbles from the jar, is it more likely that she will have 2 red marbles than that she will have one marble of each color?

1) \(y \leq 8\)
2) \(y \geq 4\)

OA B

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Manhattan Prep

A jar contains 8 red marbles and \(y\) white marbles. If Joan takes 2 random marbles from the jar, is it more likely that she will have 2 red marbles than that she will have one marble of each color?

1) \(y \leq 8\)
2) \(y \geq 4\)

Test the THRESHOLDS and try EXTREMES.
Here, the THRESHOLDS are y=8 and y=4.

Statement 1: y≤8
Case 1: 8 white marbles, 8 red marbles
P(RR) = 8/16 * 7/15 = 7/30.

P(one of each color):
P(RW) = 8/16 * 8/15 = 8/30.
Since RW can be reversed to WR, we multiply by 2:
2(8/30) = 16/30.

In this case, P(RR) < P(one of each color).

Case 2: 0 yellow marbles, 8 red marbles
Here, P(RR) = 1 and P(one of each color) = 0.
In this case, P(RR) > P(one of each color).
INSUFFICIENT.

Statement 2: y≥4
Case 1: 4 white marbles, 8 red marbles
P(RR) = 8/12 * 7/11 = 14/33.

P(one of each color):
P(RW) = 8/12 * 4/11 = 8/33.
Since RW can be reversed to WR, we multiply by 2:
2(8/33) = 16/33.

In this case, P(RR) < P(one of each color).

When y=4, P(RR) < P(one of each color).
When y=8, P(RR) < P(one of each color).
Increasing the number of yellow marbles beyond y=8 will only DECREASE the likelihood of getting RR.
Thus, P(RR) < P(one of each color).
SUFFICIENT.

The correct answer is B.