Question

In​ November, the rain in a certain valley tends to fall in storms of several​ days' duration. The unconditional probability of rain on any given day of the month is 0.500. But the probability of rain on a day that follows a rainy day is ​0.900, and the probability of rain on a day following a nonrainy day is 0.200. Find the probability of rain on November 1 and​ 2, but not on November 3.

Answer 1

Let R denote rain and N denote no rain. The probability of rain on November 1 is 0.500. The probability of rain on November 2 depends on whether it rained on November 1.

If it rained on November 1, then the probability of rain on November 2 is 0.900.

If it did not rain on November 1, then the probability of rain on November 2 is 0.200.

So, we have two cases:

Case 1: Rain on November 1

P(Rain on Nov. 1) * P(Rain on Nov. 2 | Rain on Nov. 1) * P(No Rain on Nov. 3 | Rain on Nov. 2)
= 0.500 * 0.900 * 0.500 = 0.225

Case 2: No Rain on November 1

P(No Rain on Nov. 1) * P(Rain on Nov. 2 | No Rain on Nov. 1) * P(No Rain on Nov. 3 | Rain on Nov. 2)
= 0.500 * 0.200 * 0.900 = 0.090

The probability of rain on November 1 and 2, but not on November 3, is the sum of the probabilities from the two cases:

0.225 + 0.090 = 0.315

Therefore, the probability of rain on November 1 and 2, but not on November 3, is 0.315.