Question

Assume that you have placed temperature sensors in different locations in the US. These sensors are set to automatically text you, each day, the low temperature for that day. Unfortunately, you have forgotten whether you placed a specific sensor S in DFW or in Minneapolis (but you are sure you placed it in one of those two places). The probability that you placed sensor S in DFW is 20%. The probability of getting a daily low temperature of 40 degrees or less is 20% in DFW and 80% in Minneapolis. The probability of a daily low for any day is conditionally independent of the daily low for any other day, given the location of the sensor. The sensor stays at a single place throughout your observations, and it cannot change places from day to day (it is stationary).

a) If the first text you got from sensor S indicates a daily low above 40 degrees, what is the probability that the sensor is placed in DFW?

b) If the first text you got from sensor S indicates a daily low above 40 degrees, what is the probability that the second text also indicates a daily low above 40 degrees?

c) What is the probability that the first three texts all indicate daily lows above 40 degrees?

Answer 1

Given,

Probability that sensor is in DFW = P(DFW) = 0.2

Probability that sensor is in Minneapolis = P(Minneapolis) = 1 - P(DFW) = 1 - 0.2 = 0.8

Probability of getting a daily low temperature of 40 degrees or less in DFW = P(Low | DFW) = 0.2

Probability of getting a daily low temperature of 40 degrees or less in Minneapolis = P(Low | Minneapolis) = 0.8

By law of total probability,

P(Low) = P(DFW) P(Low | DFW) + P(Minneapolis) P(Low | Minneapolis) = 0.2 * 0.2 + 0.8 * 0.8 = 0.68

a)

If the first text you got from sensor S indicates a daily low above 40 degrees, the probability that the sensor is placed in DFW

= P(DFW | Low) = P(Low | DFW) P(DFW) / P(Low) (By Bayes theorem)

= 0.2 * 0.2 / 0.68

= 0.05882353

b)

Since the probability of a daily low for any day is conditionally independent of the daily low for any other day,  probability that the second text also indicates a daily low above 40 degrees = P(Low) = 0.68

c)

Probability that the first three texts all indicate daily lows above 40 degrees = 0.68 * 0.68 * 0.68

= 0.314432