A committee of 4 students is selected at random from a group consisting of 8 girls and 4 boys. Given that there is at least one boy in the committee, the probability that there are exactly 2 boys in the committee is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{168}{425} \)
This is a conditional probability. Let F be "at least one boy" and E be "exactly 2 boys". Every committee with exactly 2 boys has at least one boy, so \( E\cap F=E \). "At least one boy" is counted through its complement, the all-girl committees.
\( \displaystyle \text{Total committees: }^{12}C_{4}=495 \)
\( \displaystyle \text{No boy (all girls): }^{8}C_{4}=70 \)
\( \displaystyle n(F)=495-70=425 \)
\( \displaystyle \text{Exactly 2 boys: }^{4}C_{2}\times ^{8}C_{2}=6\times 28=168 \)
\( \displaystyle P(E|F)=\frac{n(E\cap F)}{n(F)}=\frac{168}{425} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{85}{99} \) — Incorrect. \( \frac{85}{99}=\frac{425}{495} \) is P(F), the probability of at least one boy.
Option (3) \( \displaystyle\frac{56}{165} \) — Incorrect. \( \frac{56}{165}=\frac{168}{495} \) is the unconditional probability of exactly 2 boys; the condition reduces the sample space to 425 committees.
Option (4) \( \displaystyle\frac{257}{425} \) — Incorrect. \( \frac{257}{425}=1-\frac{168}{425} \) is the probability of 1, 3 or 4 boys, given at least one boy.
NCERT Exemplar:
Probability
