Principle of Inclusion-Exclusion

I was reading an article from Prof.Brian Alspach's poker page LINK. I was rolling along until I encountered the Principle of Inclusion-Exclusion. I looked it up but still couldn't really get a feel for it.

Could anyone explain it in layman's terms to a pretty good math skills/Non-Mathematician type

[pzhon]

Inclusion-Exclusion is a tool for counting the objects satisfying complicated conditions based on the ability to count objects satisfying simple conditions.

It is a generalization of the fact that the number of object satisfying at least one of two conditions (complicated) is the number of objects satisfying the first condition (simple) plus the number of objects satisfying the second condition (simple) minus the number of objects satisfying both conditions (we hope this is simple).

For example, if you roll a red die and a green die, what is the probability that you get at least one 6? The probability that the red die is a 6 is 1/6. The probability that the green die is a 6 is 1/6. The probability that both dice are 6s is 1/36. The probbaility that at least one is a 6 is 1/6+1/6-1/36 = 11/36.

Inclusion-Exclusion generalizes this to multiple conditions. We'd like to know the probability that at least one condition is satisfied, and it is often easier for us to determine the probability that any particular set is satisfied.

P(at least one) = Sum P(particular 1) - Sum P(particular 2) + Sum P(particular 3) - Sum P(particular 4) + ...

For example, what is the probability that at least 1 die out of 3 is a 6?

P(at least one 6) =

P(first is a 6)+P(second is a 6)+P(third is a 6)
-P(first and second are 6s)-P(second and third are 6s)-P(first and third are 6s)
+P(all three are 6s).

= 1/6+1/6+1/6-1/36-1/36-1/36+1/216
= 91/216.

See this BruceZ post for another explanation, and a nontrivial application to flush draws.

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For example, if you roll a red die and a green die, what is the probability that you get at least one 6? The probability that the red die is a 6 is 1/6. The probability that the green die is a 6 is 1/6. The probability that both dice are 6s is 1/36. The probbaility that at least one is a 6 is 1/6+1/6-1/36 = 11/36.


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It's amazing what a good example can do. Thanks. I'm gonna read that BruceZ thread.