Bayes' rule problem

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 any other day, given the location of the sensor. The sensor stays at a single place throughout your observations, it CANNOT move from Sahara to Maine between one day and another. If the sensor was in Sahara on day 1, it will stay in Sahara forever, and likewise if the sensor was in Maine on day 1, it will stay in Maine for ever.

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

Part b: If the first e-mail you got from sensor S indicates a daily high under 80 degrees, what is the probability that the second e-mail also indicates a daily high under 80 degrees?

Part c: What is the probability that the first three e-mails all indicate daily highs under 80 degrees?

Is the following answers correct? part a : 0.296 part b : 0.018 part c : 0.00246

Answer 1

Part a is ok, but part b is not.

a:

A priori probability of sensor placement: Maine, 5%, Sahara 95%

Let L = event temperature recorded $< 80$ degrees, M = sensor placed at Maine

$P(L|M) = 0.8, P(L|M^c) = 0.1$

Then $P(M|L) = P(M)P(L|M)/[P(M)P(L|M) + P(M^c)P(L|M^c] $

$= \dfrac{0.05\times 0.8} {0.05\times 0.8 + 0.95\times 0.1} \approx 0.2963$

b

Posterior probability$1$ of sensor placement: Maine, 29.63%, Sahara 70.37%

By the law of total probability, $P(L) = 0.2963\times 0.8 + 0.7037\times 0.1 \approx 0.3101$

c

The intent of this part of the question isn't clear to me, whether it is to be based only on prior probabilities, or whether it means P(third reading low | first two are low)