Probability
Probability
Allen Star Batch
Grade 12
Question:
A student appears for tests I, II, and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II, III are $p$, $q$ and $\frac{1}{2}$, respectively. If the probability that the student is successful is $\frac{1}{2}$, then: (Assuming his performance in tests are independent).
$p = 1, q = 0$
$p = 2/3, q = 1/2$
$p = 3/5, q = 2/3$
There are infinitely many values of $p$ and $q$
Step-by-Step Solution
Key Concept: Use independence to expand the union of success events into mutually exclusive intersections, then set equal to success probability.
Let $A$, $B$, $C$ be success events in tests I, II, III respectively. The student is successful if at least one of the three tests is passed: $P(\text{success}) = P[(A \cap B \cap C)^c \cup \ldots] = P(A \cap B \cap C^c) + P(A \cap B^c \cap C) + P(A^c \cap B \cap C) + P(A \cap B \cap C) = pq(1-r) + p(1-q)r + (1-p)qr + pqr$. Setting this equal to $\frac{1}{2}[pq + p(1-q) + q] = \frac{1}{2}p(1+q)$ and solving gives $p(1+q) = 1$.
Correct Answer: 1,2,3,4