Question

Awnik repeatedly plays a game that has a probability of winning of $\dfrac{1}{3}.$ The outcomes of the games are independent. What is the expected value of the number of games he will play until he has both won and lost at least once?

$\dfrac{5}{2}$

$3$

$\dfrac{16}{5}$

$\dfrac{7}{2}$

$\dfrac{15}{4}$

Solution:

The first game produces one outcome. If it was a win (probability $\tfrac{1}{3}$ ), the expected wait for a loss is $\tfrac{1}{2/3} = \tfrac{3}{2};$ if it was a loss (probability $\tfrac{2}{3}$ ), the expected wait for a win is $\tfrac{1}{1/3} = 3.$ So the expected total is $1 + \tfrac{1}{3}\cdot\tfrac{3}{2} + \tfrac{2}{3}\cdot 3 = 1 + \tfrac{1}{2} + 2 = \tfrac{7}{2}.$

Thus, the correct answer is D.

2025 AMC 12B Problem 18

Solution:

Problem 18 in Other Years