Question

Elena is planning to go to the grocery store today and wants to know whether it will be raining. In the city she lives in, the probability that it will rain on any given day (including today) is equal to 15/365, which is approximately 0.0411. The weather forecast predicts that there will be rain today, however, the weather forecast is not always right. Given that it has actually rained, the probability that the weather forecast correctly predicted it is equal to 0.85. Given that it has not actually rained, the probability that the weather forecast incorrectly predicted it is equal to 0.15. What is the probability that it will actually rain today, given that the weather forecast predicts that there will be rain?

a. 0.73%

b. 81.51%

c. 19.5%

d. 15%

e. 3.49%

Answer 1

Let us first denote the evntes

B1->it will rain on given day

B2->it will not rain on given day

P(B1)=0.0411 P(B2)=1-P(B1)=0.9589

A1:- weather forecast prediction is true

A2:- weather forecast prediction is false

P(A1|B1)=0.85 P(A2|B2)=0.15

From Bayes theorem ;

The conditional probability that it will rain today given that weather forecast prediction is true is given by below formula

P(B1|A1)=

P(B1)*P(A1|B1)/(P(B1)*P(A1|B1) + P(B2)*P(A2|B2)

=0.0411*0.85/((0.0411*0.85)+(0.9589*0.15))

=0.034935/(0.034935+0.143835)

=0.034935/0.17877

= 0.195419

Thus there will be 19.5% probablity that it will rain today given that the weather forecast is correctly predicted