Question

For a given day, temperature can be recorded as hot, mild and cold while weather can be sunny or cloudy. The probability that the temperature is hot, mild and cold are.15, .55 and .30, respectively. Probability that the weather is sunny if the temperature is hot is.67, if the temperature is recorded mild is .36, and if the temperature is cold is 33 (a)st all probabilities given in this problem, expressed in terms of events that you define (b) What is the probability that it is a sunny day? e) Given that the wateproahlty that thmild cold? (d) Given that the weather is cloudy, what is the probability that the temperature is hot? mild? cold? (e) Is the cloudy weather independent of hot weather?

Answer 1

a) Let H, M, C, S and D indicate that a day is hot, mild, cold, sunny and cloudy respectively

P(H) = 0.15

P(M) = 0.55

P(C) = 0.30

P(S | H) = 0.67

P(D | H) = 1 - 0.67 = 0.33

P(S | M) = 0.36

P(D | M) = 1 - 0.36 = 0.64

P(S | C) = 0.33

P(D | C) = 1 - 0.33 = 0.67

b) P(S) = P(H)xP(S| H) + P(M)xP(S|M) + P(C)xP(S | C)

= 0.15x0.67 + 0.55x0.36 + 0.30x0.33

= 0.3975

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

P(H I S) = 0.15x0.67/0.3975

= 0.253

P(M | S) = 0.55x0.36 / 0.3975

= 0.498

P(C | S) = 0.3x0.33/0.3975

= 0.249

d) P(D) = 1 - 0.3975

= 0.6025

P(H I D) = 0.15x0.33 / 0.6025

= 0.082

P(M | D) = 0.55x0.64 / 0.6025

= 0.584

P(C | D) = 0.30x0.67/0.6025

= 0.334

e) if D and H are independent, P(D) x P(H) must be equal to P(D and H)

P(D) x P(H) = 0.6025 x 0.15 = 0.090

P(D and H) = 0.15x0.33 = 0.0495

So, D and H are not independent