##### Example on Conditional Probablility | MIT Assignment

Most mornings, Victor checks the weather report before deciding whether to carry an umbrella. If the forecast is “rain,” the probability of actually having rain that day is 80%. On the other hand, if the forecast is “no rain,” the probability of it actually raining is equal to 10%. During fall and winter the forecast is “rain” 70% of the time and during summer and spring it is 20%.

A) One day, Victor missed the forecast and it rained. What is the probability that the forecast was “rain” if it was during the winter? What is the probability that the forecast was “rain” if it was during the summer?

The tree representation during the winter can be drawn as the following:

Let A be the event that the forecast was “Rain,”

let B be the event that it rained, and let p be the probability that the forecast says “Rain.”

If it is in the winter, p = 0.7 and

$P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{(0.8)(0.7)}{(0.8)(0.7)+(0.1)(0.3)}=\frac{56}{59}$

Similarly, if it is in the summer, p = 0.2 and

$P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{(0.8)(0.2)}{(0.8)(0.2)+(0.1)(0.8)}=\frac{2}{3}$