# Probability Question

**URL:** <https://boards.straightdope.com/t/probability-question/344359>\
**Category:** Factual Questions\
**Created:** [February 13, 2006, 11:04pm UTC](https://boards.straightdope.com/t/probability-question/344359 "2006-02-13T23:04:05Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\
**Post date:** [February 13, 2006, 11:04pm UTC](https://boards.straightdope.com/t/probability-question/344359/1 "2006-02-13T23:04:05Z")

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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-

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**Author:** ![Omphaloskeptic](https://avatars.discourse-cdn.com/v4/letter/o/bcef8e/32.png) [@Omphaloskeptic](https://boards.straightdope.com/u/Omphaloskeptic)\
**Post date:** [February 13, 2006, 11:20pm UTC](https://boards.straightdope.com/t/probability-question/344359/2 "2006-02-13T23:20:38Z")

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> [@Frylock](#):
>
> 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

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

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**Author:** ![tim314](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tim314/32/3468_2.png) [@tim314](https://boards.straightdope.com/u/tim314)\
**Post date:** [February 14, 2006, 1:15am UTC](https://boards.straightdope.com/t/probability-question/344359/3 "2006-02-14T01:15:00Z")

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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.

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [February 14, 2006, 2:57am UTC](https://boards.straightdope.com/t/probability-question/344359/4 "2006-02-14T02:57:22Z")

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> [@Frylock](#):
>
> 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.

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

```auto

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?