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Prove (3.1) for a nonuniform sample space. Hints: Remember that the probability of an event is the sum of the probabilities of the sample points favorable to it. Using Figure 3.1, let the points in A but not in AB have probabilities p1, p2, ... pn, the points in have probabilities pn+1, pn+2, .... + pn+k, and the points in B but not in AB have probabilities pn+k+1, pn+k+2, ....pn+k+l. Find each of the probabilities in (3.1) in terms of the 鈥檚 and show that you then have an identity.

Short Answer

Expert verified

It is proved that (3.1), P(AB) = P(A). PA(B).

Step by step solution

01

To find the sum of probabilities

Let p1, p2, ... pn be the probabilities of points in A but not in AB.

Assume that pn+1, pn+2, ....pn+k be the probabilities of points in AB and pn+k+1, pn+k+2, ....pn+k+lbe the probabilities of points in B but not in AB.

Therefore, the sum of probabilities of points in is given by,

p(AB) = pn+1+ ....pn+k.

Also, the sum of probabilities of points in A but not in AB is given by,

p(A) = p1 + .... + pn + pn+1+ .... + pn+k

02

To prove the result 3.1

We can write from the above two equations,

pA(B) = pn+1 + ....pn+k/p1 + .... + pn + pn+1 + .... + pn+k

.

Since, it given that A happened (considered only the points in A) and taking the sum of probabilities of points in B (which is equal to AB since A happened ).

Hence, we get,

P(AB) = P(A). PA(B)

Hence, the proof of result (3.1) is done.

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