GATE Suitability Test | Test 1 | Question: 8

Question

The chance of rainfall in the afternoon is twice that in the morning and there is no chance of rain happening at both morning and afternoon. If it rained for $3$ days, what is the probability that there was rainfall in the morning on all three days? (rounded to two decimal places)

Comments

I have approached the problem in this way:

P(rain on 1 day) = P(morning) + P(afternoon) – P(both time)

Let P(morning) = x then P(afternoon) = 2x and given P(both) = 0

P(rain on 1 day) = x + 2x – 0 = 3x

P(3 day rain) = P(day 1 rain) * P(day 2 rain) *P(day 3 rain) = 3x * 3x * 3x = 27x^3

P(3 day morning | 3 day rain) = P(3 day morning ^ 3 day rain) / P(3 day rain) [from conditional probability definition]

P(3 day morning ^ 3 day rain) = P(3 day morning) [Since if it rains on morning then it will not rain on afternoon] = x*x*x = x^3

=> P(3 day morning | 3 day rain) = x^3/ (27*x^3) => 1/27

Is my working correct?

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is there some mistake in question?, in place of evening there should be afternoon, so the final answer should be 1/27.

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but rain on 1st day, 2nd day or 3rd day are independent events, why are they multiplied?