91 Suppose that in Vancouver BC the probability that a rainy

91. Suppose that in Vancouver, B.C., the probability that a rainy fall day is followed by a rainy day is 0.80 and the probability that a sunny day is followed by a rainy clay is 0.6. Find the probabilities that a rainy fall day is followed by (a) a rainy day, a sunny day, and another rainy day; (b) two sunny days and then a rainy day; (c) two rainy days and two sunny days; (d) rain two days later; [Hint: In part (c) use formula of Exercise 19.]

Solution

let

P(A) = probability of rainy day followed by rainy day = 0.8

P(B) = probabaility of sunny day followed by rainy day = 0.6

P(C) = probability of sunny day followed by sunny day = 1- P(B) = 0.4

P(D) = probability of rainy day followed by sunny day = 1- P(A) = 0.2

a)

probability of rainy day followed by rainy day,sunny day and another rainy day

= P(A) * P(D) * P(B)

= 0.8 * 0.2 * 0.6

= 0.096

b)

Probability of rainy day followed by two sunny days and then a rainy day

= P(D) * P(C) * P(B)

= 0.2 * 0.4 * 0.6

= 0.048

c)

Probability of rainy day followed by two rainy days and two sunny days

= P(A) * P(A) * P(D) * P(C)

= 0.8 * 0.8 * 0.2 * 0.4

= 0.0512

d)

Probability of rain two days later

= P(D) * P(B) + P(A) * P(A)

= 0.2 * 0.6 + 0.8 * 0.8

= 0.76