06/07/2020 (2011: National Sprint Round, Problem 23)

Q: In 2010, the probability of Mark sinking a free throw was twice what it was in 2009, and yet the probability of him sinking exactly two out of three free throws was the same in both years. What was the probability of Mark sinking a free throw in 2010 given that the probability was greater than zero? Express your answer as a common fraction.

A: 6/7

Let's say that "M" is Mark's probability of sinking a free throw in 2009. Then we get the following equation:

M² * (1 - M) = (2M)² * (1 - 2M), which simplifies to:

M² - M³ = -8M³ + 4M² -->

7M³ - 3M² = 0 -->

7M - 3 = 0 -->

7M = 3 -->

M = 3/7

We are trying to find Mark's probability of sinking a free throw in 2010 (we solved for 2009), so we multiply 3/7 by 2 and get 6/7.