Bayesian network probability problem. Rainy on 3rd day given it rains on 1st day

Question

Given the above bayesian network constructed in order to predict rain during successive days and the conditional probability tables. What would be the probability of P( Rain3∣ Rain1 )?

My approach:

There are two possible ways for P( Rain3∣ Rain1 )

1. Either it rains on day 1, not day 2 and day 3 (in the table 0.350)
2. Or rains on day 1, day 2 and day 3 (in the table 0.612)

Let's denote these as random variables A and B respectively.

Therefore shouldn't P( Rain3∣ Rain1 ) = P(A∪B)

And since A and B are given in the table as A=0.350, B=0.612 then P(A∪B) = 0.350 + 0.612 = 0.962

Answer 1

Almost had it...

Rains on day 1, not day 2 and day 3 ...

• Given that it rains on day 1, the prob of not raining on day 2 is $1-0.586 = 0.414$ $$P(R_3 = T \cap R_2 = F | R_1 = T) = (0.414)(0.350) = 0.145 $$ Rains on day 1, day 2 and day 3 ...
• Given that it rains on day 1, the prob of raining on day 2 is $0.586$ $$P(R_3 = T \cap R_2 = T | R_1 = T) =0.586(0.612) = 0.359$$

Finally... $$P(R_3 = T | R_1 = T) =0.359+0.145 = 0.504 $$