1. There are three identical and independent temperature sensors that will trigger in: 

- 90% of the cases where the temperature is high

- 5% of the cases where the temperature is nominal

- 1% of the cases where the temperature is low.

The probability of high temperature is 20%, nominal temperature is 70%, and low temperature is 10%. Describe a Bayesian network and corresponding queries for computing the following:

(a) Probability that the first sensor will trigger given that the other two sensors have also triggered

(b) Probability that the temperature is high given that all three sensors have triggered

(c) Probability that the temperature is high given that at least one sensor has triggered

2. A few weeks after inseminating a cow, we have three possible tests to confirm pregnancy. The first is a scanning test (S) that has a false positive of 1% and a false negative of 10%. The second is a blood test (B) that detects progesterone with a false positive of 10% and a false negative of 30%. The third test is a urine test (U) that also detects progesterone with a false positive of 10% and a false negative of 20%. The probability of a detectable progesterone level is 90% given pregnancy and 1% given no pregnancy. The probability that insemination will impregnate a cow is 87%.

Suppose now that we inseminate a cow, wait for a few weeks, and then perform the three tests, which all come out negative. Using a Bayesian model determine the probability that the cow is pregnant?