2020 CIME II Problems/Problem 6

Problem

An infinite number of buckets, labeled $1$ , $2$ , $3$ , $\ldots$ , lie in a line. A red ball, a green ball, and a blue ball are each tossed into a bucket, such that for each ball, the probability the ball lands in bucket $k$ is $2^{-k}$ . Given that all three balls land in the same bucket $B$ and that $B$ is even, then the expected value of $B$ can be expressed as $\tfrac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

The probability that all three balls land in box $2n$ is $\frac{1}{64^n}$ . This means that the probability that the three balls land in the same even box is $\frac{1}{64} + \frac{1}{64^2} + \ldots = \frac{1}{63}$ . This means that the probability that all three balls land in box $2n$ $\emph{given that they land in the same even box}$

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2020 CIME II Problems/Problem 6

Problem

Solution