A group has 30 men and 20 women. Of these, 10 men and 8 women wear glasses. A person is chosen at random from the group. If the person chosen wears glasses, then the probability that the person is a woman is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{4}{9}\)
Since the person is known to wear glasses, only the people who wear glasses are possible. This reduced sample space has
\( 10 + 8 = 18 \) people
Of these 18 people, 8 are women.
\( \displaystyle P(\text{woman} \mid \text{glasses}) = \frac{8}{18} \)
\( \displaystyle = \frac{4}{9} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{4}{25}\) — Incorrect. This is \(\frac{8}{50}\), the probability that the person is a woman who wears glasses, out of the whole group.
Option (2) \(\displaystyle\frac{2}{5}\) — Incorrect. This is \(\frac{8}{20}\), the probability that a woman wears glasses; it conditions on the wrong event.
Option (4) \(\displaystyle\frac{5}{9}\) — Incorrect. This is \(\frac{10}{18}\), the probability that the person is a man.