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Probability Theory [5 P]

a)
Prove the conditionalized version of Bayes' rule:

$\displaystyle P(A\vert B,C) = \frac{P(B\vert A,C) \cdot P(A\vert C)}{P(B\vert C)} $

b)
Show that the statement

$\displaystyle P(A,B\vert C) = P(A\vert C) P(B\vert C)$

is equivalent to either of the statements

$\displaystyle P(A\vert B,C) = P(A\vert C)  $and$\displaystyle  P(B\vert A,C) = P(B\vert C)$

c)
After your early checkup, the doctor has bad news and good news. The bad news is that you tested positive for a serious disease and that the test is 99% accurate (i.e., the probability of testing positive when you do have the disease is 0.99, as is the probability of testing negative when you don't have the disease). The good news is that this is a rare disease, striking only 1 in 10000 people of your age. Why is it good news that the disease is rare? What are the chances that you actually have the disease?

d)
Suppose you are a witness to a nighttime hit-and-run accident involving a taxi in Athens. All taxis in Athens are blue or green. You swear, under oath, that the taxi was blue. Extensive testing shows that, under the dim lighting conditions, discrimination between blue and green it 75% reliable. Is it possible to calculate the most likely color for the taxi? (Hint: distinguish carefully between the proposition that the taxi is blue and the proposition that it appears blue.)

What about now, given that 9 out of 10 Athenian taxis are green?

e)
Three prisoners, $ A$ , $ B$ , and $ C$ , are locked in their cells. It is common knowledge that one of them will be executed the next day and the others pardoned. Only the governor knows which one will be executed. Prisoner $ A$ asks the guard a favor: "Please ask the governor who will be executed, and then take a message to one of my friends $ B$ or $ C$ to let him know that he will be pardoned in the morning." The guard agrees, and comes back later and tells $ A$ that he gave the pardon message to $ B$ .

What are $ A$ 's chances of being executed, given this information?


next up previous
Next: Decision Theory I [1* Up: NNA_Exercises_2009 Previous: Exercises
Haeusler Stefan 2010-01-19