Question

You are a meteorologist that places temperature sensors all of the world, and you set them up so that they automatically e-mail you, each day, the high temperature for that day. Unfortunately, you have forgotten whether you placed a certain sensor S in Maine or in the Sahara desert (but you are sure you placed it in one of those two places) . The probability that you placed sensor S in Maine is 5%. The probability of getting a daily high temperature of 80 degrees or more is 20% in Maine and 90% in Sahara. Assume that probability of a daily high for any day is conditionally independent of the daily high for the previous day, given the location of the sensor. Part a: If the first e-mail you got from sensor S indicates a daily high over 80 degrees, what is the probability that the sensor is placed in Maine

Answer 1

If the first e-mail you got from sensor S indicates a daily high over 80 degrees, the probability that the sensor is placed in Maine = 0.01156

Explanation:

Let the probability of placing the sensor S at Maines be P(M) = 0.05

Probabilty of placing the sensor S in the Sahara desert = P(S) = 1 - P(M) = 1 - 0.05 = 0.95 (this is because the meteorologist is confident that the sensor is at either of the two places.

Let the probability of a high temperature of 80 degrees or more = P(T)

Probability of the sensor picking up temperature of 80 degrees or more given that it is in Maine = P(T|M) = 20% = 0.20

Probability of the sensor picking up temperature of 80 degrees or more given that it is in the Sahara desert = P(T|S) = 90% = 0.90

From the definition of rhe conditional probability

P(A|B) = P(A n B) ÷ P(B)

P(A n B) = P(A|B) × P(B)

If the first e-mail you got from sensor S indicates a daily high over 80 degrees, what is the probability that the sensor is placed in Maine?

The required probability = P(M|T)

P(M|T) = P(M n T) ÷ P(T)

P(T) = P(M n T) + P(S n T)

P(M n T) = P(T|M) × P(M) = 0.20 × 0.05 = 0.010

P(S n T) = P(T|S) × P(S) = 0.90 × 0.95 = 0.855

P(T) = 0 010 + 0.855 = 0.865

P(M|T) = P(M n T) ÷ P(T)

P(M|T) = (0.010/0.865) = 0.01156

Hope this Helps!!!