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<blockquote data-quote="invitro" data-source="post: 215618" data-attributes="member: 446"><p><strong>Coupon Collector's Problem</strong></p><p></p><p></p><p>I thought someone said somewhere that you didn't get duplicates, but a couple of you say you can... I've got I think 7 of them with no dupes. Anyway, if you can get duplicates, the expected number of tokens needed to have all 26 is:</p><p></p><p>the expected number needed to get the 1st token, which is 26/26 = 1, PLUS</p><p>the expected number needed to get the 2nd different token, once you have one, which is (26 possibilities)/(25 successes) = 26/25, PLUS</p><p>the expected number needed to get the 3rd different token, once you have two different ones, which is 26/24, PLUS</p><p>the expected number needed to get the 4th different token, once you have three different ones, which is 26/23, PLUS</p><p>...</p><p>the expected number needed to get the 25th different token, once you have 24 different ones, which is 26/2 = 13, PLUS</p><p>the expected number needed to get the 26th different token, once you have 25 different ones, which is 26/1 = 26</p><p></p><p>= 26 * (1/26 + 1/25 + 1/24 + ... + 1/2 + 1/1)</p><p></p><p>= 100.2, to one decimal place. I got an average value of 99-101 with a simulation, so I'm pretty sure this is correct.</p><p></p><p>There's a nice write-up on this problem at <a href="https://en.wikipedia.org/wiki/Coupon_collector%27s_problem" target="_blank">https://en.wikipedia.org/wiki/Coupon_collector's_problem</a>.</p></blockquote><p></p>
[QUOTE="invitro, post: 215618, member: 446"] [b]Coupon Collector's Problem[/b] I thought someone said somewhere that you didn't get duplicates, but a couple of you say you can... I've got I think 7 of them with no dupes. Anyway, if you can get duplicates, the expected number of tokens needed to have all 26 is: the expected number needed to get the 1st token, which is 26/26 = 1, PLUS the expected number needed to get the 2nd different token, once you have one, which is (26 possibilities)/(25 successes) = 26/25, PLUS the expected number needed to get the 3rd different token, once you have two different ones, which is 26/24, PLUS the expected number needed to get the 4th different token, once you have three different ones, which is 26/23, PLUS ... the expected number needed to get the 25th different token, once you have 24 different ones, which is 26/2 = 13, PLUS the expected number needed to get the 26th different token, once you have 25 different ones, which is 26/1 = 26 = 26 * (1/26 + 1/25 + 1/24 + ... + 1/2 + 1/1) = 100.2, to one decimal place. I got an average value of 99-101 with a simulation, so I'm pretty sure this is correct. There's a nice write-up on this problem at [URL]https://en.wikipedia.org/wiki/Coupon_collector%27s_problem[/URL]. [/QUOTE]
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