The probability that it rains in a town on a Monday is 0.4. If it rains on Monday, the probability that it also rains on Tuesday is 0.7; if it does not rain on Monday, the probability that it rains on Tuesday is 0.2. The probability that it rains on exactly one of these two days is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0.24
Let M be the event “rain on Monday” and T the event “rain on Tuesday”. Then P(M) = 0.4, P(T|M) = 0.7 and P(T|M′) = 0.2. Exactly one rainy day can happen in two separate ways.
Rain on Monday only, by the multiplication theorem:
\(P(M\cap T') = P(M)\,P(T'|M)\)
\(P(M\cap T') = 0.4\times 0.3 = 0.12\)
Rain on Tuesday only:
\(P(M'\cap T) = P(M')\,P(T|M')\)
\(P(M'\cap T) = 0.6\times 0.2 = 0.12\)
The two ways cannot happen together, so
\(P = 0.12+0.12 = 0.24\)
Why the other options are incorrect
Option (1) 0.28 — Incorrect. This is \(0.4\times 0.7\), the probability of rain on both days.
Option (3) 0.40 — Incorrect. This is \(0.28+0.12\), the probability of rain on Tuesday.
Option (4) 0.48 — Incorrect. This is \(0.6\times 0.8\), the probability of no rain on either day.