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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