Unformatted text preview: Proposition 3.1. If E and F are independent, then E and F c are independent. Proof. P ( E ∩ F c ) = P ( E ) P ( E ∩ F ) = P ( E ) P ( E ) P ( F ) = P ( E )[1 P ( F )] = P ( E ) P ( F c ) . We say E , F , and G are independent if E and F are independent, E and G are independent, F and G are independent, and P ( E ∩ F ∩ G ) = P ( E ) P ( F ) P ( G ). Example. Suppose you roll two dice, E is that the sum is 7, F that the first is a 4, and G that the second is a 3. E and F are independent, as are E and G and F and G , but E,F and G are not. Example. What is the probability that exactly 3 threes will show if you roll 10 dice? Answer. The probability that the 1st, 2nd, and 4th dice will show a three and the other 7 will not is 1 6 3 5 6 7 . Independence is used here: the probability is 1 6 1 6 5 6 1 6 5 6 ··· 5 6 . The probability that the 4th, 5th, and 6th dice will show a three and the other 7 will not is the same thing. So to answer our original question, we take 1 6 3 5 6 7...
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This note was uploaded on 12/29/2011 for the course MATH 317 taught by Professor Wen during the Spring '09 term at SUNY Stony Brook.
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