elemprob-fall2010-page18

# elemprob-fall2010-page18 - ± 5 2 ²± 3 2 ² ± 8 4 ² If...

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we are looking at the number of successes in 365 “trials.” In other words, we are looking at a binomial. The above proposition says that this can be approximated very well by a Poisson. If instead of days, we look at minutes, the approximation is even better. 7 Hypergeometric distributions Suppose we have an urn with 5 red marbles and 3 green marbles. We pick 4 marbles without replacement and ask, what is the probability we have 2 red marbles and 2 green marbles? We could also pick 4 marbles with replacement and get a different answer. There are 8 4 ways of choosing 4 marbles out of 8 total, all equally likely. There are 5 2 ways of choosing 2 red marbles out of the 5 red ones and
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Unformatted text preview: ± 5 2 ²± 3 2 ² ± 8 4 ² . If there are m of one type, n of another, and we want N total to be chosen, with k of hte ﬁrst type, the probability becomes ± m k ²± n N-k ² ± m + n N ² . A random variable with these probabilities ( m,n,N are parameters, k the possible values) is said to have a hypergeometric distribution. An example. What is the probability that in a poker hand (5 cards out of 52) we get exactly 4 of a kind? Answer: The probability of 4 aces and 1 king is ± 4 4 ²± 4 1 ² . ± 52 5 ² . The probability of 4 jacks and one 3 is the same. There are 13 ways to pick the 18...
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