hmwk_9 - n people chosen at random, there are at least two...

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Discrete Structures Oct 24 2011 Assignment 9: Due at beginning of class Monday, Oct 31 Prof. Hopcroft *******Note: this homework is subject to the style guide on the website. Points will be deducted for homeworks not following the guidelines.******** *******Please print out and staple the grade sheet to the back of your homework!****** 1. Prove that if E and F are events, which happen with probability P ( E ) and P ( F ) respectively, then: P ( E F ) P ( E ) + P ( F ) - 1 . 2. Let E 1 ,E 2 ,...,E n be events. Prove by induction that P ( E 1 E 2 ∩··· ∩ E n ) P ( E 1 ) + P ( E 2 ) + ··· + P ( E n ) - n + 1 . 3. Let E and F be independent events. Prove that, event not E , denoted as E , and event not F , denoted as F , are independent events. 4. (a) What is the probability that two people chosen at random were born on the same day of the week? (b) What is the probability in a group of
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Unformatted text preview: n people chosen at random, there are at least two born on the same day of the week? (c) How many people chosen at random are needed to make the probability greater than 1 2 that there are at least two people born in the same month of the year? 5. Suppose we know 8% of players use steroids and 92% do not. We have a test that returns positive if the test thinks the player used steroids. When a player on steroids takes the test, they have a 96% chance of testing positive. When a player not on steroids takes the test, they have a 9% chance of testing positive. Now, we test a player and the test returns positive. What is the probability the player was using steroids? 1...
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