Question

The probability of men getting a certain disease is $\frac{1}{2}$ and that of women getting the same disease is $\frac{1}{5}$. The blood test that identifies the disease gives the correct result with probability $\frac{4}{5}$. Suppose a person is chosen at random from a group of $30$ males and $20$ females, and the blood test of that person is found to be positive. What is the probability that the chosen person is a man?

• A. $\frac{75}{107}$
• B. $\frac{3}{5}$
• C. $\frac{15}{19}$
• D. $\frac{3}{10}$

Answer 1

The correct option is A $\frac{75}{107}$


Probability that a man is chosen,
P$=\frac{\left(1\right)+\left(3\right)}{\left(1\right)+\left(3\right)+\left(5\right)+\left(7\right)}=\frac{\frac{3}{5}\times \frac{1}{2}\times \frac{4}{5}+\frac{3}{5}\times \frac{1}{2}\times \frac{1}{5}}{\frac{3}{5}\times \frac{1}{2}\times \frac{4}{5}+\frac{3}{5}\times \frac{1}{2}\times \frac{1}{5}+\frac{2}{5}\times \frac{1}{5}\times \frac{4}{5}+\frac{2}{5}\times \frac{4}{5}\times \frac{1}{5}}$

P $=\frac{\frac{3}{10}}{\frac{3}{10}+\frac{16}{125}}=\frac{75}{107}$