# 2019 AMC 8 Problems/Problem 15

## Problem 15

On a beach $50$ people are wearing sunglasses and $35$ people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is $\frac{2}{5}$. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?

$\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}$

## Solution 1

The number of people wearing caps and sunglasses is $\frac{2}{5}\cdot35=14$. So then, 14 people out of the 50 people wearing sunglasses also have caps.

$\frac{14}{50}=\boxed{\textbf{(B)}\frac{7}{25}}$

## Solution 2

Let $A$ be the event that a randomly selected person is wearing sunglasses, and let $B$ be the event that a randomly selected person is wearing a cap. We can write $P(A \cap B)$ in two ways: $P(A)P(B|A)$ or $P(B)P(A|B)$. Suppose there are $t$ people in total. Then $$P(A) = \frac{50}{t}$$ and $$P(B) = \frac{35}{t}.$$ Additionally, we know that the probability that someone is wearing sunglasses given that they wear a cap is $\frac{2}{5}$, so $P(A|B) = \frac{2}{5}$. We let $P(B|A)$, which is the quantity we want to find, be equal to $x$. Substituting in, we get $$\frac{50}{t} \cdot x = \frac{35}{t} \cdot \frac{2}{5}$$$$\implies 50x = 35 \cdot \frac{2}{5}$$ $$\implies 50x = 14$$ $$\implies x = \frac{14}{50}$$ $$= \boxed{\textbf{(B)}~\frac{7}{25}}$$

Note: This solution makes use of the dependent events probability formula, $P(A \cap B) = P(A)P(B|A)$, where $P(B|A)$ represents the probability that $B$ occurs given that $A$ has already occurred and $P(A \cup B)$ represents the probability of both $A$ and $B$ happening.

~ cxsmi

## Video Solution by Math-X (First fully understand the problem!!!)

~Math-X

### Solution Explained

https://youtu.be/gOZOCFNXMhE ~ The Learning Royal

~ pi_is_3.14

~ MathEx

Another video

-Happytwin

### Video Solution

Solution detailing how to solve the problem: https://www.youtube.com/watch?v=omRgmX7KXOg&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=16

~savannahsolver

### Video Solution (MOST EFFICIENT+ CREATIVE THINKING!!!)

~Education, the Study of Everything

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