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2019 AMC 12B Problem 13

Problem 13 of 25IntermediateProbability & StatisticsProblem-Solving Techniques

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 kk is 2−k2^{-k} for k=1,k=1, 2,2, 3,3, …\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 ∑k=1∞(2−k)2=∑k=1∞4−k=141−14=13. \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 1−132=13. \dfrac{1-\tfrac13}{2}=\dfrac13. Thus, C is the correct answer.
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Tagged: basic probability · symmetry · complementary probability

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