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

The conditional probability that a person is male, given that he has colorblind is0.9524

The conditional probability that a person is male, given that he has colorblind is 0.9756

Step 1 Given Information of 1

Given that  5 percent of men and 0.25 percent of women are color blind. A color-blind person is chosen at random 

Also given that there are an equal number of males and females.

So, the probability for males is $0.5$ and

probability for a female is$0.5$

We have to find he conditional probability that a person is male, given that he has colorblind 

 

Step 2 Tree Diagram of 1

Diagram Tree

Step 3 Explanation of 1

Let $E$ denote the event that the person has colorblind$, F$ denote the event that the selected person is female, and$M$ denote the event that the selected person is male.

Thus,

$P(M\mid E)=0.05$, $P(F\mid E)=0.0025,\text{ and }P\left(M\right)=P\left(F\right)=0.5$

The conditional probability that a person is male, given that he has colorblind is,

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

$=\frac{0.05\times 0.5}{(0.05\times 0.5)+(0.0025\times 0.5)}$

$=\frac{0.025}{0.02625}$

$=0.9524$

Step 4 Final Answer of 1

The conditional probability that a person is male, given that he has colorblind is 0.9524

Step 5 Given Information of 2

Given that  5 percent of men and 0.25 percent of women are color blind. A color-blind person is chosen at random 

Also given that there are an equal number of males and females.

So, the probability for males is 0.5  and probability for a female is0.5

We have to find the conditional probability that a person is male, given that he has colorblind  

Step 6 Diagram Tree of 2

Suppose the population consisted of twice as many males as females.

Then, the tree diagram is shown in below:

Step 7 Explanation of 2

So,

$P(M\mid E)=0.05,P(F\mid E)=0.0025,P\left(M\right)=\frac{2}{3},\text{ and }P\left(F\right)=\frac{1}{3}$

The conditional probability that person is male, given that he has colorblind is,

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

$=\frac{0.05\times \frac{2}{3}}{\left(0.05\times \frac{2}{3}\right)+\left(0.0025\times \frac{1}{3}\right)}$

$=\frac{0.033333}{0.034167}$

=0.9756

Step 8 Final Answer of 2

The conditional probability that a person is male, given that he has colorblind  is0.9756