Difference between revisions of "2019 AMC 8 Problems/Problem 15"

Latest revision as of 20:57, 5 June 2025

Problem

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 EzLx

https://youtu.be/SkDf39Cg5z4

~EzLx CookeMonster SirCookies

Video Solution

https://youtu.be/6xNkyDgIhEE?t=250pih-jsm

@@ Line 1: / Line 1: @@
-==Problem 15==
+==Problem==
 On a beach <math>50</math> people are wearing sunglasses and <math>35</math> 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 <math>\frac{2}{5}</math>. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?
@@ Line 20: / Line 20: @@
 ~ cxsmi
-==Video Solutions==
+==Video Solution by EzLx==
+https://youtu.be/SkDf39Cg5z4
-==Video Solution by Math-X (First fully understand the problem!!!)==
+~EzLx CookeMonster SirCookies
-https://youtu.be/IgpayYB48C4?si=V_SNrrp17pztxbQG&t=4518
-~Math-X
-===Solution Explained===
-https://youtu.be/gOZOCFNXMhE ~ The Learning Royal
-===Video Solution by OmegaLearn===
-https://youtu.be/6xNkyDgIhEE?t=252
-~ pi_is_3.14
 ===Video Solution===
-https://www.youtube.com/watch?v=gKlYlAiBzrs
+https://youtu.be/6xNkyDgIhEE?t=250pih-jsm
-~ MathEx
-https://www.youtube.com/watch?v=afMsUqER13c
-Another video
-https://youtu.be/37UWNaltvQo
--Happytwin
-=== Video Solution ===
-Solution detailing how to solve the problem: https://www.youtube.com/watch?v=omRgmX7KXOg&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=16
-===Video Solution===
-https://youtu.be/9nbaMSAQCNU
-~savannahsolver
-===Video Solution (MOST EFFICIENT+ CREATIVE THINKING!!!)===
-https://youtu.be/cK_mfkZ_rYM
-~Education, the Study of Everything
-===Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)===
-https://youtu.be/Xm4ZGND9WoY
-~Hayabusa1
 ==See Also==
@@ Line 71: / Line 32: @@
 {{MAA Notice}}
+[[Category:Introductory Probability Problems]]

2019 AMC 8 (Problems • Answer Key • Resources)
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