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Combinatorics - Probability
union sets probability

probability with union sets




We have:
A ? B = (X ? A) ? (A ? B) ? (Y ? B)   (1)
Then:
P(A ? B) = P(X ? A) + P(A ? B) + P(Y ? B)   (1')

A = (X ? A) ? (A ? B)   (2)
B = (Y ? B) ? (A ? B)   (3)

P(A) = P(X ? A) + P(A ? B)   (2')
P(B) = P(Y ? B) + P(A ? B)   (3')

P(X ? A) = P(A) - P(A ? B)    (2")
P(Y ? B) = P(B) - P(A ? B)   (3")


P(A ? B) = P(X ? A) + P(A ? B) + P(Y ? B) (1")
= P(A) - P(A ? B) + P(A ? B) + P(B) - P(A ? B)
= P(A) + P(B) - P(A ? B)

P(A ? B) = P(A) + P(B) - P(A ? B)


P(A ? B) = P(A) + P(B) - P(A ? B)





  
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