8.11 ( ) Consider the example of the car fuel system shown in Figure 8.21, and suppose that instead of observing the state of the fuel gauge G directly, the gauge is seen by the driver D who reports to us the reading on the gauge. This report is either that the gauge shows full D = 1or that it shows empty D = 0. Our driver is a bit unreliable, as expressed through the following probabilities p(D = 1|G = 1) = 0.9 (8.105)

p(D = 0|G = 0) = 0.9. (8.106)

Suppose that the driver tells us that the fuel gauge shows empty, in other words that we observe D = 0. Evaluate the probability that the tank is empty given only this observation. Similarly, evaluate the corresponding probability given also the observation that the battery is flat, and note that this second probability is lower.

Discuss the intuition behind this result, and relate the result to Figure 8.54.