2. What does it mean for one event C to cause another event E for example, smok-

Question

2. What does it mean for one event C to cause another event E for example, smok- ing (C) to cause cancer (E)? There is a long history in philosophy, statistics, and the sciences of trying to clearly analyze the concept of a cause. One tradition says that causes raise the probability of their effects; we may write this symbolically is PEIC)> P(E). (a) Does equation (1) imply that P(CIE)>PC)? If so, prove it. If not, give (b) Another way to formulate a probabilistic theory of causation is to say that a counter example. P(EIC) > PE C) Show that equation (1) implies equation (2) c)Let C be the drop in the level of mercury in a barometer and let E be a storm. Briefly describe why this leads to a problem with using equation (1) (or equation (2) as a theory of causation (d) Let A, C, and E be events. If P(EIA nC) = P(EIC), then C is said to screen A off from E. Suppose that P(EnC) >0. Show that screening off is equivalent to saying that P(AnEIC)- P(AIC)P(EIC). What does this latter equation say in terms of independence? (e) Now let A be a the drop in the level of mercury in a barometer, E be a storm, and C be a drop in atmospheric pressure. Does the result from part (d) help fix the problem suggested in part (c)?

Explanation / Answer

a. P(C|E) = P(E|C). P(C)/P(E)

As P(E|C)>P(E), P(C|E) > P(C)

b. We have P(E) = P(C).P(E|C) + P(C').P(E|C')

So, P(E) = (1-P(C')).P(E|C) + P(C').P(E|C')

P(E) - P(E|C) = P(C')(P(E|C') - P(E|C))

From 1, P(E) - P(E|C) < 0 and P(C')>0,

P(E|C') - P(E|C) < 0

So, P(E|C) > P(E|C')

c. The problem with causation here is that event C does not cause event E but event E causes event C

d. P(AE|C) = P(AEC)/ P(C) = P(AEC)/ P(AC). P(AC)/P(C)

P(AE|C) = P(E|AC) . P(A|C)

AS P(E|AC) = P(E|C),

P(AE|C) = P(E|C) . P(A|C)

Hence A and E are independent given event C has occured

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