Question

Suppose I have faith in my local weather reporter who says that the chance of rain tomorrow is 80 percent. Moreover, my friend, the seasoned sailor, tells me that whenever it rains the next day, the appearance of the sky visible this evening only appears about 10 percent of the time. Whenever there is a clear day, he adds, the present appearance of the sky occurs 60 percent of the time the evening before. Assume I trust my friends judgement, what would be my belief in the chance of rain tomorrow, given the appearance of the sky this evening?

Answer 1

Let

• R =event that it rains tomorrow
• NR= event that tomorrow is a clear day
• S=event of the appearance of the sky this evening

We know the following probabilities

the chance of rain tomorrow is 80 percent. this is same as the probability that it rains tomorrow is 0.80

P(R) = 0.80

the probability that tomorrow is a clear day is

P(NR) = 1-P(R) = 1-0.80 = 0.20

whenever it rains the next day, the appearance of the sky visible this evening only appears about 10 percent of the time

is same as the conditional probability of the appearance of the sky this evening given that it rains tomorrow is 0.10

Whenever there is a clear day, the present appearance of the sky occurs 60 percent of the time the evening before.

This is same as the conditional probability of the appearance of the sky this evening given that tomorrow is a clear day is 0.60

the chance of rain tomorrow, given the appearance of the sky this evening is same as

the conditional probability of rain tomorrow (Event R) , given the appearance of the sky this evening (Event S)

ans:  my belief in the chance of rain tomorrow, given the appearance of the sky this evening is 40%