Question

There is a $40\%$ chance of rain on Saturday and a $30\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find $a + b.$

Answer

Solution:

Let $p$ be the probability that it rains on Sunday given a dry Saturday; given a rainy Saturday it is $2p.$ The overall Sunday chance gives $0.4(2p) + 0.6p = 0.3,$ so $1.4p = 0.3$ and $p = \frac{3}{14}.$

The weekend is completely dry exactly when Saturday is dry and then Sunday is dry: $\frac{3}{5}\left(1 - \frac{3}{14}\right) = \frac{3}{5} \cdot \frac{11}{14} = \frac{33}{70}.$ So the probability of rain on at least one day is $1 - \frac{33}{70} = \frac{37}{70},$ which is in lowest terms, and $a + b = 37 + 70 = 107.$

2016 AIME II Problem 2

Solution:

Problem 2 in Other Years