Each midwinter day in Minneapolis the weather may be classif

Each mid-winter day in Minneapolis the weather may be classified as warm, cold, or brutally cold. The probability that a winter day is warm is 0.20, and the probability that it is cold is 0.7. In addition, on each day it may snow or not snow. The probability of snow on warm, cold, and brutally cold days is 0.4, 0.2, and 0.1, respectively.

(a) What is the probability that it will snow on any day?
(b) If it does snow, what is the probability that it is a warm day?

Solution

As per the problem,
Probability that a winter day is warm, P(W) = 0.20
Probability that a winter day is cold, P(C) = 0.70
Probability that a winter day is brutally cold, P(B) = 0.10         (Since P(W)+ P(C)+ P(B) = 1)
Probability that it snows, given that the day is warm, P(S|W) = 0.40
Probability that it snows, given that the day is cold, P(S|C) = 0.20
Probability that it snows, given that the day is brutally cold, P(S|B) = 0.10

a. The probability that it will snow on any day, P(S) =
     P(SW) + P(SC) + P(SB)
= P(S|W) * P(W) + P(S|C) * P(C) + P(S|B) * P(B)                     (Since P(SW) = P(S|W) * P(W))
  = 0.40 * 0.20 + 0.20 * 0.70 + 0.10 * 0.10
  = 0.23

b. The probability that it is a warm day, given that it snows, P(W|S) =
     P(WS) / P(S)
= P(S|W) * P(W) / P(S)
= 0.40 * 0.20 / 0.23                                                                       (P(S) has been calculated from part a)
= 0.35