elemprob-fall2010-page4 - i th toss turns up heads, then we...

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2 Independence A and B are independent if P ( A B ) = P ( A ) P ( B ) . Proposition 2.1 If A and B are independent, then A c and B are indepen- dent. Proof. P ( A c B ) = P ( B ) - P ( A B ) = P ( B ) - P ( A ) P ( B ) = P ( B )(1 - P ( A )) . Students often make the mistake of thinking that every event is indepen- dent of itself. If A is independent of itself, then P ( A ) = P ( A A ) = P ( A ) P ( A ) , and this happens if and only if P ( A ) is zero or one. If we have a sequence of events A 1 ,A 2 ... , then they are independent if P ( A i 1 ∩ ··· ∩ A i n ) = P ( A i 1 ) ··· P ( A i n ) for every subset i 1 ,...,i n of distinct elements. Suppose we toss a coin repetitively. If A i is the event that the
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Unformatted text preview: i th toss turns up heads, then we say the tosses are independent if the events A i are independent. Suppose we toss a coin repetitively and independently, and let A i be as before. Let us assume that the probability of a heads is p . Let X be the number of the toss at which we rst get a heads. Then X being equal to n is the event A c 1 A c 2 A c n-1 A n . By independence, the probability of this is (1-p ) n-1 p. 4...
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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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