Question

Consider the following events:

R - it will rain tomorrow (same as on any given day of the year)

Rc - it will not rain tomorrow

F - tomorow's forcast says it will rain

Here is what we know from the past:

P(R) = 0.25 (25% of the days were rainy)

P( F | R ) = 0.85    (85% of the rainy days were corectly predicted)

P( F | Rc ) = 0.20   (20% of the not rainy days had rainy forecasts)

Question:

What is the probability that it will rain tomorrow, given the forecast calls for rain?
P( R | F ) = ?

Answer 1

Given P(F|R) = P(F and R)/P(R) = 0.85 => P(F and R) = 0.85*P(R) = 0.85*0.25 = 0.2125

P(F|Rc) = P(F and Rc)/P(Rc) = 0.20 But P(Rc) = 1 - P(R) => P(F and Rc) = 0.20*(1-P(R)) = 0.20*(1-0.25) = 0.20*0.75 = 0.15

We know that, P(F) = P(F and R) + P(F and Rc) (from venn diagram rules) = 0.2125+0.15 = 0.3625

So, P(R|F) = P( R and F)/P(F) = 0.2125/0.3625 = 0.586