Two thirds of the students in a class are boys and the rest are girls. It is known that the probability of a girl getting a first class is 0.25 and that of a boy getting a first class is 0.28, then the probability that a student chosen at random will get first class marks in the subject is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0.27
A student is either a boy or a girl, so the theorem of total probability applies.
Let B = the student is a boy, G = the student is a girl, F = the student gets first class.
\( \displaystyle P(B)=\frac{2}{3},\quad P(G)=\frac{1}{3} \)
\( \displaystyle P(F|B)=0.28,\quad P(F|G)=0.25 \)
\( \displaystyle P(F)=P(B)P(F|B)+P(G)P(F|G) \)
\( \displaystyle =\frac{2}{3}(0.28)+\frac{1}{3}(0.25) \)
\( \displaystyle =\frac{0.56+0.25}{3} \)
\( \displaystyle =\frac{0.81}{3} \)
\( \displaystyle =0.27 \)
Shortcut: P(F) is a weighted average of 0.28 and 0.25, so it must lie between them. Only 0.27 does.