Probability Question

Factual Questions
1

Tell me where I’m going wrong, please.

If I recall correctly, given two events A and B whose probailities are independent, the probability of “A and B” is P(A) * P(B) and the probability of “A or B or both” is P(A) + P(B) - P(A and B)

Now, take the following two statements:

P: A occurs, and (either B occurs, or C occurs, or both occur.)
Q: Either (A and B both occur,) or (A and C both occur,) or both of these situations occur.

Now, for at least some probability assignments to A, B, and C, the probabilities of these two events (meaning events P and Q) differ. For example, if P(A) = 2/3, P(B) = 1/3, and P© = 1/3, then P§ is 10/27, while P(Q) is 32/81.

Yet, I can’t think of a way to picture either of these occuring without the other also occuring. In fact, A^(BvC) is logically equivalent to (A^B)v(A^C) if I remember correctly.

If, for some R and S, there is no way for R to occur without S also occuring, and there is no way for S to occur without R also occuring, then as far as I can tell, R and S must have the same probability.

But P and Q are such a pair, and yet do not have the same probability.

Where’s my error?

Thanks!

-FrL-

2

Are the events “A and B both occur” and “A and C both occur” independent, if A, B, and C are all independent events?

3

In other words, the probability of A^B and A^C both occuring is just the probability of A and B and C occuring. In otherwords, it’s just P(A)*P(B)*P©
It isn’t P(A^B) * P(A^C). = P(A)*P(B)*P(A)*P©

Thus, when calculating the probability of Q, you want:
P(A^B) + P(A^C) - P((A^B)^(A^C)) = P(A)*P(B) + P(A)*P© - P(A)*P(B)*P©

With this correction, the probability of Q is the same as the probability of P.

4

If I can butcher a bit of boolean algebra here, take a look at this:


p = a(bc' + b'c + bc)
   = a(bc' + c)
   = abc' + ac

q = abc' + ab'c + abc
   = abc' + ac

Do you understand where the original expressions came from?