Probability
Conditional probability — exactly one scorer
MJAT_TS8_P1
Grade 12
Question:
Students $S_1,S_2,S_3$ agree: $P(E_1)=\frac{1}{2}$ (no one scores $>75\%$), $P(E_2)=\frac{1}{5}$ ($S_3$ scores $>75\%$), $P(E_3)=\frac{1}{10}$ ($S_3$ scores $>75\%$ given at least two score $>75\%$), $P(E_4)=\frac{1}{4}$ (exactly one scores $>75\%$). The probability that only $S_3$ scores above 75\% marks is:
A) $\dfrac{1}{40}$
B) $\dfrac{3}{40}$
C) $\dfrac{7}{40}$
D) $\dfrac{9}{40}$
Step-by-Step Solution
Key Concept: From $P(E_3)$: $P(S_3$ scores, at least 2 score$)=\frac{1}{10}\cdot P(\text{at least 2})$. $P(\text{at least 2})=1-P(E_1)-P(E_4)=1-\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$. So $P(S_3\cap\text{at least 2})=\frac{1}{10}\cdot\frac{1}{4}=\frac{1}{40}$.
Probability $=\mathbf{\dfrac{7}{40}}$. Answer: C.
Correct Answer: C