Banana StickersDate: 05/29/97 at 15:11:52 From: nathan & jim Subject: Banana - probability You collect banana stickers and each banana sticker has a different letter on it (a-z). If you collect n {n: n > 26} banana stickers, what is the probability that you'll have all 26 letters? Date: 06/01/97 at 20:17:23 From: Doctor Anthony Subject: Re: Banana - probability We can tackle this by what is known as the urn model. You have 26 urns and n balls, and the balls are distributed at random among the 26 urns. What is the probability that there are no empty urns? The number of ways of distributing n numbered balls into 26 numbered boxes such that no box is empty is given by T(n,26). The total number of ways of distributing the balls without restriction is 26^n, so the required probability is T(n,26) -------- 26^n T(n,26) will be the coefficient of x^n/n! in the expansion of [e^x - 1]^26 = e^(26x) - C(26,1)e^(25x) + C(26,2)e^(24x) - .. The term in x^n/n! will be (x^n/n!).[26^n - C(26,1)25^n + C(26,2)24^n - .....] and the term in square brackets is T(n,26) This will be a very lengthy calculation but it is necessary to ensure that our probability is calculated from equiprobabe events. You might also want to take a look at this related item from our archive: Collecting a Set of Coupons http://mathforum.org/library/drmath/view/56657.html -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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