HMMT February, 2023

Combinatorics Difficulty 5.6 AIME, harder Prove it United States

Problem:

There are $800$ marbles in a bag. Each marble is colored with one of $100$ colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops.

Suppose Anna has not stopped after drawing $699$ marbles. Compute the probability that she stops immediately after drawing the $700$ th marble.

Solution

Solution:

In order to not stop after $699$ marbles, the last $101$ marbles must consist of $2$ marbles of one color, and one marble from each other color. Since each of these marbles is equally likely to be the next to be drawn, and we stop after drawing the next marble as long as it's not one of the two of the same color, the desired probability is simply $\frac{99}{101}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.