AMC 8 2019 Problem 15

AMC8 2019 年第 15 题

概率★★★☆☆

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

\displaystyle \frac{2}{5}

. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?

在一个沙滩上，50 人戴着太阳镜，35 人戴着帽子，有些人同时戴着太阳镜和帽子。如果从戴着帽子的人当中随机选一人，那么这个人同时也戴着太阳镜的概率是

\displaystyle \frac{2}{5}

。相反，如果从戴着太阳镜的人当中随机选择一人，那么这个人同时也戴着帽子的概率是多少？

A. $\displaystyle \frac{14}{85}$
B. $\displaystyle \frac{7}{25}$
C. $\displaystyle \frac{2}{5}$
D. $\displaystyle \frac{4}{7}$
E. $\displaystyle \frac{7}{10}$

正确答案 · Correct Answer

解析 · Solution

Solution 1

The number of people wearing caps and sunglasses is

\displaystyle \frac{2}{5} \cdot 35=14

. So then, 14 people out of the 50 people wearing sunglasses also have caps.

\displaystyle \frac{14}{50} = \frac{7}{25}

. 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

\displaystyle \frac{2}{5}

, so

\displaystyle 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}

= \frac{7}{25}

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