.
(1) probability of rain on day one is 50%, and
(2) if it rain on day 1 probability of rain on day 2 is 20%, and
(3) if it do not rain on day 1 probability or rain on day 2 is also 20%
find the probability of it rained on first day given it rain on second day.
~~~~~~~~~~~~~~~~~
Let A be the event "there is a rain on day 1".
Let B be the event "there is a rain on day 2".
We are given
(1) P(A) = 0.5;
(2) P(B|A) = 0.2;
(3) P(B|notA) = 0.2.
+---------------------------------+
| They want you find P(A|B). |
+---------------------------------+
From (1) and (2), we have P(A ∩ B) = P(B|A)*P(A) = 0.2*0.5 = 0.1.
From (1) and (3), we have P(notA ∩ B) = P(B|notA)*P(notA) = 0.2*0.5 = 0.1.
Hence, P(B) = P(A ∩ B) + P(notA ∩ B) = 0.1 + 0.1 = 0.2.
It implies P(A|B) = P(A ∩ B)/P(B) = 0.1/0.2 = 1/2 = 0.5 = 50%.
So, the solution is formally complete and the answer is
+-----------------------------------------------+
| "the probability of it rained on first day |
| given it rains on second day is 0.5". |
+-----------------------------------------------+
It is a formal solution, but the informal solution is OBVIOUS:
Indeed, from the given part, the probability that it rains the second day
is 20% independently on what was happened on the 1st day.
So, the fact that "it is given 20% probability of the rain in the 2nd day" does not add any new information.
Hence, the only condition (1) works to predict the weather at the 1st day,
and it says that it is 50% probability of rain, giving the answer to the problem's question.
Solved and explained - - - both formally and informally.
