elemprob-fall2010-page3

# elemprob-fall2010-page3 - Here is an example Suppose we...

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Here is an example. Suppose we have 100 students, of which 50 take French, 30 take Spanish, and 20 take both. How many take at least one of the two languages? 30 = 50 - 20 take just French, 10 = 30 - 20 take just Spanish, and 20 take both, so 70 = 30 + 10 + 20 take at least one. Proposition 1.3 If A B , then P ( A ) P ( B ) . Proof. P ( B ) = P ( A ) + P ( B A c ) P ( A ) . Proposition 1.4 Suppose A i A . This means A 1 A 2 ⊂ ··· and A = i =1 A i . Then P ( A ) = lim i →∞ P ( A i ) . Proof. Let B 1 = A 1 , B 2 = A 2 A c 1 , B 3 = A 3 A c 2 , and so on. Then the B i are disjoint, their union is A , and the union of the ﬁrst n of them is A n . So by the deﬁnition of inﬁnite sum,
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## This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.

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