2019 AMC 12B Problem 13

Problem 13 of 25IntermediateCounting & Probability

A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin

k

2^{-k}

for

k=1,

2,

3,

\ldots

What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?

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Solution

The probability the balls land in the same bin is

\begin{gathered} \sum_{k=1}^\infty \left(2^{-k}\right)^2=\sum_{k=1}^\infty 4^{-k} \\ =\dfrac{\frac{1}{4}}{1-\frac{1}{4}}=\dfrac13. \end{gathered}

By symmetry, the red ball being higher and the green ball being higher are equally likely, so each has probability

\dfrac{1-\tfrac13}{2}=\dfrac13.

Thus, C is the correct answer.

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