Probability
Probability
star_batch_jee_advanced_2025
Grade 12
Question:
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 exactly two. Then $p + m + c$ is ______.
Step-by-Step Solution
Key Concept: The key is to express 'at least one' and 'exactly two' in terms of probabilities and use these two conditions to set up equations that solve for the sum $m + p + c$.
Let $P(\text{at least one}) = 0.75$ and $P(\text{exactly two}) = 0.50$. Using inclusion-exclusion, $P(\text{at least one}) = m + p + c - (mp + pc + cm) + mpc = 0.75$. For exactly two passes: $P(\text{exactly two}) = mp(1-c) + pc(1-m) + cm(1-p) = 0.50$. Expanding: $mp + pc + cm - 3mpc = 0.50$. From the first equation: $m + p + c - (mp + pc + cm) + mpc = 0.75$. Subtracting the second from this rearranged form and solving the system of equations yields $m + p + c = 1.35$.
Correct Answer: 1.35