Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (𝐻), medium (𝑀) and low (𝐿)

Q. Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (𝐻), medium (𝑀) and low (𝐿). Let 𝑃(𝐻_𝐺) denote the probability that Guwahati has high temperature. Similarly, 𝑃(𝑀_𝐺) and 𝑃(𝐿_𝐺) denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use 𝑃(𝐻_𝐷), 𝑃(𝑀_𝐷) and 𝑃(𝐿_𝐷) for Delhi.

The following table gives the conditional probabilities for Delhi’s temperature given Guwahati’s temperature.

Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature (𝐻_𝐺) then the probability of Delhi also having a high temperature (𝐻_𝐷) is 0.40; i.e., 𝑃(𝐻_𝐷|𝐻_𝐺) = 0.40. Similarly, the next two entries are 𝑃(𝑀_𝐷|𝐻_𝐺) = 0.48 and 𝑃(𝐿_𝐷|𝐻_𝐺) = 0.12. Similarly for the other rows.

If it is known that 𝑃(𝐻_𝐺) = 0.2, 𝑃(𝑀_𝐺) = 0.5, and 𝑃(𝐿_𝐺) = 0.3, then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is 

Ans: 0.6015

Sol:

P(H_D) = P(H_D ∣ H_G) × P(H_G) + P(H_D ∣ M_G) × P(M_G) + P(H_D ∣ L_G) × P(L_G)
P(H_D) = 0.4 × 0.2 + 0.1 × 0.5 + 0.01 × 0.3 = 0.133

Therefore,
P(H_G ∣ H_D) = P(H_G ∩ H_D) / P(H_D)
P(H_G ∣ H_D) = P(H_D ∣ H_G) × P(H_G) / P(H_D)
P(H_G ∣ H_D) = 0.40 × 0.2 / P(H_D) = 0.08 / 0.133 = 0.6015