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Combinatorics Difficulty 4.8 AIME Find the answer Canada

Angie has a jar that contains 2 red marbles, 2 blue marbles, and no other marbles. She randomly draws 2 marbles from the jar. If the marbles are the same colour, she discards one and puts the other back into the jar. If the marbles are different colours, she discards the red marble and puts the blue marble back into the jar. She repeats this process a total of three times. What is the probability that the remaining marble is red?

Pick one

Solution

Solution 1

Let the letter $R$ represent a red marble, and the letter $B$ represent a blue marble.

On her first draw, Angie may draw $RR$ , $RB$ or $BB$ .

Case 1: Angie draws $RR$ or $RB$ on her first draw

If Angie draws $RR$ or $RB$ on her first draw, then she discards the $R$ and the three remaining marbles in the jar are $RBB$ .

On her second draw, Angie may draw $RB$ or $BB$ .

If she draws $RB$ , then she discards the $R$ and the two remaining marbles in the jar are $BB$ .

Since there are no red marbles remaining, it is not possible for the final marble to be red in this case.

If on her second draw Angie instead draws $BB$ , then she discards a $B$ and the two remaining marbles in the jar are $RB$ .

When these are both drawn on her third draw, the $R$ is discarded and the final marble is blue.

Again in this case it is not possible for the final marble to be red.

Thus, if Angie draws $RR$ or $RB$ on her first draw, the probability that the final marble is red is zero.

Case 2: Angie draws $BB$ on her first draw

If Angie draws $BB$ on her first draw, then she discards a $B$ and the three remaining marbles in the jar are $RRB$ .

On her second draw, Angie may draw $RR$ or $RB$ .

If she draws $RR$ or $RB$ , then she discards one $R$ and the two remaining marbles in the jar are $RB$ .

When these are both drawn on her third draw, the $R$ is discarded and the final marble is blue.

In this case it is not possible for the final marble to be red.

Thus, if Angie draws $BB$ on her first draw, the probability that the final marble is red is zero.

Therefore, under the given conditions of drawing and discarding marbles, the probability that Angie’s last remaining marble is red is zero.

Solution 2

Let the letter $R$ represent a red marble, and the letter $B$ represent a blue marble.

If the final remaining marble is $R$ , then the last two marbles must include at least one $R$ .

That is, the last two marbles must be $RB$ or $RR$ .

If the last two marbles are $RB$ , then when they are drawn from the jar, the $R$ is discarded and the $B$ would remain.

Thus it is not possible for the final marble to be $R$ if the final two marbles are $RB$ .

So the final remaining marble is $R$ only if the final two marbles are $RR$ .

If the final two marbles are $RR$ , then the last three marbles are $BRR$ (since there are only two $R$ s in the jar at the beginning).

However, if the final three marbles are $BRR$ , then when Angie draws two of these marbles from the jar, at least one of the marbles must be $R$ and therefore one $R$ will be discarded leaving $BR$ as the final two marbles in the jar.

That is, it is not possible for the final two marbles in the jar to be $RR$ .

The only possibility that the final remaining marble is $R$ occurs when the final two marbles are $RR$ , but this is not possible.

Therefore, under the given conditions of drawing and discarding marbles, the probability that Angie’s last remaining marble is red is zero.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.