Question

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 $k$ is $2^{-k}$ 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?

$\dfrac{1}{4}$

$\dfrac{2}{7}$

$\dfrac{1}{3}$

$\dfrac{3}{8}$

$\dfrac{3}{7}$

Solution:

The probability the balls land in the same bin is $\sum_{k=1}^\infty \left(2^{-k}\right)^2=\sum_{k=1}^\infty 4^{-k}=\dfrac{1/4}{1-1/4}=\dfrac13.$

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.

2019 AMC 12B Problem 13

Solution:

Problem 13 in Other Years