Suppose that 5 percent of men and .25 percent of women are color blind

# Suppose that 5 percent of men and .25 percent of women are color blind

J
240 points

Suppose that 5 percent of men and .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?

conditional probability
bayes theorem

8.7k points

Let C be the event that a person is colorblind, M be the event that a person is male, and W the event that they are female. We are given that,

$$P(C|M) = 0.5$$

and

$$P(C|W) = 0.0025$$

What is the probability of this person being male?

We use Bayes rule to find,

$$P(M|C) = \frac{P(C|M)P(M)}{P(C)}$$

to find $P(C)$ use the relation,

$$P(C)=P(C|M)P(M)+P(C|W)P(W)=0.05(0.5)+0.0025(0.5)=0.02625$$

and hence,

$$P(M|C)=\frac{0.05(0.5)}{0.02625}=0.9523$$

What if the population consisted of twice as many males as females

By this information we have that $P(M)=2P(F)$ or in other words $P(M)=\frac{2}{3}$ and $P(F) = \frac{1}{3}$ and so,

$$P(CB)=0.05(2/3)+0.0025(1/3)=0.03416$$

giving,

$$P(M|C)=\frac{0.05(2/3)}{0.03416}=0.9756$$

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