Q. 3.26 - A First Course in Probability

Question

 Suppose that$5$ percent of men and $0.25$ percent of women are color blind. A color-blind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females. What if the population consisted of twice as many males as females?

Step-by-Step Solution

Answer

Determine the proportions of men and women in the defined populations. Apply the Bayes formula:
$$\begin{gathered} \text{ [2ex] }{P}_1=\frac{20}{21}\left[\text{ lex }\right] \\ {P}_2=\frac{40}{41} \end{gathered}$$

Step1:Introduction

Considered events:

C - A person chosen at random is colorblind.

M -A male was chosen at random.

F - A female was chosen at random.

Given probabilities:

$\begin{array}{c}P(C\mid M)=5\mathrm{\%}\\ P(C\mid F)=0.25\mathrm{\%}\end{array}$

$P\left(M\right)=P\left(F\right)$

$\text{ II) }P\left(M\right)=2\times P\left(F\right)$

Step2: Find men and women make up the whole population

Use the Bayes formula (obtained by breaking C into C M and C F )

$P(M\mid C)=\frac{P(C\mid M)\times P\left(M\right)}{P(C\mid M)P\left(M\right)+P(C\mid F)P\left(F\right)}$

The majority of the population is made up of men and women.

$$\begin{gathered} P\left(M\right)=P\left(F\right) \\ \Rightarrow P\left(M\right)=P\left(F\right) \\ =0.5 \end{gathered}$$

$P(M\mid C)=\frac{0.05\times 0.5}{0.05\times 0.5+0.0025\times 0.5}$

$=\frac{20}{21}$

Step3: The probability of this person being male

The majority of the population is made up of men and women.

$$\begin{gathered} P\left(M\right)=2P\left(F\right)\Rightarrow P\left(M\right) \\ =\frac{2}{3}, \\ P\left(F\right)=\frac{1}{3} \end{gathered}$$

$P(M\mid C)=\frac{0.05\times \frac{2}{3}}{0.05\times \frac{2}{3}+0.0025\times \frac{1}{3}}$

$=\frac{40}{41}$