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Probability
โ€ƒThe Inclusion-Exclusion Principle

Probability

The Inclusion-Exclusion Principle

For a given ๐‘ objects, suppose that some of these objects have property ๐›ผ, and some do not. Let ๐‘(๐›ผ) denote the number having property ๐›ผ. Similarly, suppose that some of the objects have property ๐›ฝ, and some do not. Let ๐‘(๐›ฝ) denote the number having property ๐›ฝ. If there are other properties ๐›พ, ๐›ฟ, โ‹ฏ, let ๐‘(๐›พ), ๐‘(๐›ฟ), โ‹ฏ denote the number of objects having property ๐›พ, the number having property ๐›ฟ, โ‹ฏ.

Continuing the general analysis, let ๐‘(๐›ผ,๐›ฝ) denote the number of objects having both properties ๐›ผ and ๐›ฝ. Let ๐‘(๐›ผ,๐›ฝ,๐›พ) denote the number of objects having the three properties ๐›ผ, ๐›ฝ and ๐›พ. In the same way ๐‘(๐›ผ,๐›ฝ,๐›พ,๐›ฟ) denotes the number of objects having the four properties ๐›ผ, ๐›ฝ, ๐›พ and ๐›ฟ.

Therefore, the ๐‘ objects that do not have property ๐›ผ is equal to ๐‘โˆ’๐‘(๐›ผ). And the ๐‘ objects that do not have neither the property ๐›ผ and ๐›ฝ is equal to ๐‘โˆ’๐‘(๐›ผ)โˆ’๐‘(๐›ฝ)+๐‘(๐›ผ,๐›ฝ). And the ๐‘ objects that do not have the properties ๐›ผ, ๐›ฝ, and ๐›พ is equal to ๐‘โˆ’๐‘(๐›ผ)โˆ’๐‘(๐›ฝ)โˆ’๐‘(๐›พ)+๐‘(๐›ผ,๐›ฝ)+๐‘(๐›ผ,๐›พ)+๐‘(๐›ฝ,๐›พ)โˆ’๐‘(๐›ผ,๐›ฝ,๐›พ). Inclusion-exclusion principleThe number of objects having none of the properties ๐›ผ, ๐›ฝ, ๐›พ, โ‹ฏ  ๐‘ โˆ’๐‘(๐›ผ)โˆ’๐‘(๐›ฝ)โˆ’๐‘(๐›พ)โˆ’โ‹ฏ +๐‘(๐›ผ,๐›ฝ)+๐‘(๐›ผ,๐›พ)+๐‘(๐›ฝ,๐›พ)+โ‹ฏ โˆ’๐‘(๐›ผ,๐›ฝ,๐›พ)โˆ’โ‹ฏ โ‹ฎ

Consider an object, ๐‘‡, that has exactly ๐‘— of the properties, where ๐‘— is some positive integer. ๐‘‡ is counted by the term ๐‘. And โˆ’๐‘(๐›ผ)โˆ’๐‘(๐›ฝ)โˆ’๐‘(๐›พ)โˆ’โ‹ฏ object ๐‘‡ is counted ๐‘— times, or what is the same thing, ๐ถ(๐‘—,1) times. And +๐‘(๐›ผ,๐›ฝ)+๐‘(๐›ผ,๐›พ)+๐‘(๐›ฝ,๐›พ)+โ‹ฏ object ๐‘‡ is counted ๐ถ(๐‘—,2), because this is the number of terms with two of the ๐‘— properties of ๐‘‡. Similarly the numbers of terms with other combination of ๐‘— properties are ๐ถ(๐‘—,3), ๐ถ(๐‘—,4), โ‹ฏ Inclusion-exclusion principleThe number of objects having none of the ๐‘— properties 1โˆ’๐ถ(๐‘—,1)+๐ถ(๐‘—,2)โˆ’๐ถ(๐‘—,3)+๐ถ(๐‘—,4)โˆ’+โ‹ฏ=0


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ID: 190500012 Last Updated: 12/5/2019 Revision: 0 Ref:

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References

  1. B. Joseph, 1978
  2. Wheatstone, C., 1854
  3. Stroud, K.A., 2001
  4. Coolidge, J.L., 1949
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