conditional probability problem

Consider the problem of locating the mouse in the triangular pen we discussed in class. The nature of the problem is changing a bit. Rather than being in a triangular pen, he now can only travel in a tube. At any point in time he could be at one end or the other of the tube, or he could be in the middle. There is a microphone at each end of the tube that is a bit more accurate than the old microphone. We have also learned a bit more about the habits of the mouse in an attempt to help determine where he is. We now know that sometimes he is sleeping and not moving around, and the rest of the time he is moving around looking for food. Our new set of probabilistic information is given by:

Microphone quality: If the mouse is moving around in the same end of the tube as the microphone, he is heard 0.7 of the time. If he is at the other end of the tube away from the microphone, he is heard 0.05 of the time. If the mouse is moving around in the middle, either microphone will hear him 0.5 of the time. If the mouse is sleeping he rarely makes any noise. Therefore, regardless of where he is, a microphone only hears him 0.03 of the time.

If the mouse is sleeping at one time unit, the odds of him still sleeping at the next time unit are 0.8 and he will not have changed location. If the mouse was moving around at one time unit, the odds of him falling asleep and being in the same location at the next time unit are 0.3. If a mouse woke up between one time unit and the next, or was awake the entire time, he may have changed his location. If he was in the middle he stays in the same spot 0.6 of the time, and moves to either end 0.2 of the time each. If he was in one end he stays in that end 0.7 of the time, moves to the middle 0.2 of the time, and to the other corner 0.1 of the time.

In the beginning we saw that the mouse entered the left end of the tube, so we believe that there was a 0.5 chance he was still there when we started the tracking process, and 0.3 chance he was in the middle, and a 0.2 chance he had moved to the right end. We think there was a 0.9 chance that he was still awake.

a) Assume that at time unit 1 the mouse was heard only by the microphone at the right end of the tube. Compute the probabilities of his possible locations. Compute the odds of him being awake or asleep.

b) At time unit 2 the mouse was heard by both microphones. Update your location and awake probabilities.

c) At time unit 3 the mouse was not heard by either microphone. Update your location and awake probabilities.

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