Question

Most mornings, I check the weather report before deciding whether to carry an umbrella. If the forecast is “rain”, the probability of actually having rain is 70%. On the other hand, if the forecast is “no rain”, the probability of actually raining is equal to 15%. During fall and winter, the forecast is “rain”75% of the time and during summer and spring, it is 15%. a) One day, I missed the forecast and it rained. What is the probability that the forecast was “no rain” If it was during the winter? What is probability that the forecast was “rain” if it was during summer?

Answer 1

Bayes' Theorem: P(A | B) = P(A & B)/P(B)

a) In winter,

P(forecast was no rain | it rained) = P(forecast was no rain and it rained) / P(it rained)

= P(forecast was no rain and it rained) / [P(forecast was no rain and it rained) + P(forecast was rain and it rained)]

= (0.25x0.15)/(0.25x0.15 + 0.75x0.70)

= 0.0667

In summer,

P(forecast was rain | it rained) = P(forecast was rain and it rained) / P(it rained)

= P(forecast was rain and it rained) / [P(forecast was rain and it rained) + P(forecast was no rain and it rained)]

= (0.15x0.70)/(0.15x0.70 + 0.85x0.15)

= 0.4516