3. There are three rooms in a house: the kitchen, the bedroom and the living room. There is a radio transmitter outside the home that can be received in each room. A user is wearing a device that can measure the strength of the signal; however, due to noise in the sensor and in the environment, each time the device reads the signal strength, the result may contain an error. We know that the bedroom should have a reading of 50 units; the kitchen a reading of 51 units; and the living room a reading of 52 units in 50% of the cases and 53 in the other 50% of the cases. Someone has made a lot of measurements and created the following table representing the probability distribution for a given reading is in a particular room (P(Reading|Room)):

Room\Reading

<= 49

50

51

52

53

54<=

Bedroom

0.33

0.41

0.13

0.08

0.05

0

Kitchen

0.07

0.13

0.53

0.14

0.08

0.05

Living Room

0.06

0.09

0.13

0.28

0.33

0.11

a. What are the probabilities (that we are inside) for each room given that we obtain a reading of 52 (and assuming that we spend about equal amounts of time in each room - i.e., that the sample comes equally likely from any of the three rooms)? (25 points)

b. For each reading we have our best guess (highest probability) determining which room we are in. What is our chance for being wrong when the reading is 52? (20 points)

c. How would the result in B change if we knew that we are spending twice as much time in the bedroom than in the kitchen or living room (but the same in these two)? (25 points)