A group of 100 people has 40 men and 60 women. Of the men, 12 smoke, and of the women, x smoke. A person is chosen at random from the group. If the events “the person is a man” and “the person smokes” are independent, then x is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 18
Let M be the event “the person is a man” and S the event “the person smokes”. There are 12 + x smokers in all.
\( \displaystyle P(M) = \frac{40}{100},\quad P(S) = \frac{12+x}{100},\quad P(M\cap S) = \frac{12}{100} \)
For independent events, \(P(M\cap S) = P(M)P(S)\):
\( \displaystyle \frac{12}{100} = \frac{40}{100}\times\frac{12+x}{100} \)
\( 1200 = 40(12+x) \)
\( 12+x = 30 \)
\( x = 18 \)
Check: 12 of the 40 men and 18 of the 60 women smoke, that is 30% in each group, so knowing whether the person is a man does not change the chance of smoking.
Why the other options are incorrect
Option (1) 12 — Incorrect. Then 12 of the 60 women smoke, that is 20%, while 30% of the men smoke.
Option (2) 20 — Incorrect. Then \(P(M)P(S) = 0.4\times 0.32 = 0.128\), not 0.12.
Option (3) 30 — Incorrect. 30 is the total number of smokers needed, 12 + x, not x.