conditional probability

# conditional probability - Example Suppose 10 people put a...

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Example. Suppose 10 people put a key into a hat and then withdraw one randomly. What is the probability at least one person gets his/her own key? Answer. If E i is the event that the i th person gets his/her own key, we want P ( 10 i =1 E i ). One can show, either from a picture or an induction proof, that P ( 10 i =1 E i ) = i 1 P ( E i 1 ) - i 1 <i 2 P ( E i 1 E i 2 ) + i 1 <i 2 <i 3 P ( E i 1 E i 2 E i 3 ) - · · · . Now the probability that at least the 1st, 3rd, 5th, and 7th person gets his or her own key is the number of ways the 2nd, 4th, 6th, 8th, 9th, and 10th person can choose a key out of 6, namely 6!, divided by the number of ways 10 people can each choose a key, namely 10!. so P ( E 1 E 3 E 5 E 7 ) = 6! / 10!. There are 10 4 ways of selecting 4 people to have their own key out of 10, so i 1 ,i 2 ,i 3 ,i 4 P ( E i 1 E i 2 E i 3 E i 4 ) = 10 4 6! 10! = 1 4! The other terms are similar, and the answer is 1 1! - 1 2! + 1 3! - · · · - 1 10! 1 - e - 1 . 3. Conditional probability. Suppose there are 200 men, of which 100 are smokers, and 100 women, of which 20 are smokers. What is the probability that a person chosen at random will be a smoker? The answer is 120 / 300. Now. let us

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