Question

In London, half of the days have some rain. The weather forecaster is correct 2/3 of thetime, i.e., the probability that it rains, given that she has predicted rain, and the probabilitythat it does not rain, given that she has predicted that it won’t rain, are both equal to 2/3.When rain is forecast, Mr. Pickwick takes his umbrella. When rain is not forecast, he takesit with probability 1/3. Find (a) the probability that Pickwick has no umbrella, given that itrains. (b) the probability that it doesn’t rain, given that he brings his umbrella.

Answer 1

The other person is right

Explanation:

Answer 2

(a) The probability that Pickwick has no umbrella, given that it rains is 2/9

(b) The probability that it doesn’t rain, given that he brings his umbrella is 5/9

Explanation:

(a) The probability that Pickwick has no umbrella, given that it rains: P1

P1 = P (rain) * P (not umbrella)

This probability have two components, if rain forecasts or not rain forecasts:

1. When rain is forecast Mr. Pickwick takes his umbrella, so the probability of not umbrella is zero and the probability of rain for a rain forecast is 2/3 (the weather forecaster is correct 2/3 of the time)
2. When not rain forecast, he takes it with probability 1/3, so the probability of not takes it is 2/3 and the probability of rain with a not rain forecast is 1/3 (the weather forecaster is correct 2/3 of the time, so there is 1/3 probability of rain)

P1 = 2/3*0 + 1/3*2/3 = 0 + 2/9 = 2/9

The probability that Pickwick has no umbrella, given that it rains is 2/9

(b) The probability that it doesn’t rain, given that he brings his umbrella: P2

P2 = P (not rain) * P (umbrella)

This probability have two components, if rain forecasts or not rain forecasts:

1. When rain is forecast Mr. Pickwick takes his umbrella, so the probability is one and the probability of rain for a rain forecast is 2/3, so the probability of rain is 1/3 (the weather forecaster is correct 2/3 of the time, so 1/3 will be wrong)
2. When not rain forecast, he takes it with probability 1/3 and the probability of not rain with a not rain forecast is 2/3 (the weather forecaster is correct 2/3 of the time)

P1 = 1/3*1 + 2/3*1/3 = 1/3 + 2/9 = 5/9

The probability that it doesn’t rain, given that he brings his umbrella is 5/9