Question

You are going to meet a friend at the airport. Your experience tells you that the plane is late 70% of the time when it rains, but is only late 20% of the time when it doesn’t rain. The weather forecast for that morning calls for a 40% chance of rain. What is the probability that it rained, if the plane is late?

Answer 1

The probability that it rained that morning, given that the plane was late P( R | L) = 70%

Explanation:

We shall be using Baye’s theorem to solve this;

Let the event that it rains be R

Let the event that it does not rain be N

Let the event that the plane is late be L

From the question;

P( R) that morning = 40% = 0.4

P(N) that morning = 60% = 0.6

P(L | R) read as probability that the plane is late , given that it rained = 70% = 0.7

P( L | N) read as probability that the plane is late given that there is no rain = 20% = 0.2

Now, we want to calculate P ( R | L) which is read as probability that it rained given that the plane was late

What we shall use to solve this issue is the Baye’s theorem of conditional probability.

By Baye’s theorem;

P( R | L) = R) * P(R) /( P( R) * P (L | R) + P(N) * P(L | N)

Substituting the values;

P(R | L) = (0.7 * 0.4)/(0.4 * 0.7) + (0.6 * 0.2)

P(R | L) = 0.28/(0.28 + 0.12)

P( R | L) = 0.28/0.4

P( R | L) = 0.7 = 70%