IIT-JEE 1999Question 1
The probabilities that a student passes in Mathematics, Physics and Chemistry are $m$, $p$ and $c$ respectively. Of these subjects, the student has a $75\%$ chance of passing in at least one, a $50\%$ chance of passing in at least two, and a $40\%$ chance of passing in exactly two. Which of the following relations are true?
Probability
Solution
With $S_1=\sum P_i$, $S_2=\sum P_iP_j$, $S_3=pmc$: $P(\ge2)=S_2-2S_3=\frac12$ and $P(\text{exactly }2)=S_2-3S_3=\frac25$, so $S_3=\frac1{10}$, $S_2=\frac7{10}$. Then $P(\ge1)=S_1-S_2+S_3=\frac34$ gives $S_1=\frac{27}{20}$.