tag:blogger.com,1999:blog-20067416.post4009190601426159076..comments2017-01-19T04:40:28.485-05:00Comments on Recursivity: A Neat Problem on Card ArrangementsJeffrey Shallitnoreply@blogger.comBlogger8125tag:blogger.com,1999:blog-20067416.post-90821312819049751992010-01-31T19:36:17.833-05:002010-01-31T19:36:17.833-05:00Here is another interesting probability problem.<a href="http://randomprocessed.blogspot.com/2010/01/totally-nontrivial-prisoners-problem.html" rel="nofollow">Here</a> is another interesting probability problem.Takis Konstantopouloshttp://www.blogger.com/profile/14675216467783238403noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-77483366777016681522010-01-26T14:17:26.350-05:002010-01-26T14:17:26.350-05:00It worked! .4863215... But now I won't be able...It worked! .4863215... But now I won't be able to do arithmetic for a week without feeling sick.<br /><br />My idea was this: the number of shuffles of a 52 card deck in which either AK or KA appears is the same as the number of shuffles of a 53 card deck (52 + joker) in which the first card is a joker, and at least one of the following sequences appears:<br /><br />XAK<br />XKA <br />XAAK <br />XKKA <br />XAAAK <br />XKKKA <br />XAAAAK <br />XKKKKA<br /><br />Here 'X' means any of the 44 + 1 = 45 cards that are neither Ace nor King. But that number is just 1/45 times the number of shuffles of 53 in which one of the above sequence appears, and the first card is an X. So you just subtract out the 53 card shuffles (containing at least one of the above sequences) that start with an Ace or a King. Having the 'X' as a separator allows you to make easy use of the inclusion-exclusion principle.jasonhttp://www.blogger.com/profile/15972096640733349029noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-31487170502747656962010-01-25T16:24:07.277-05:002010-01-25T16:24:07.277-05:00After several days of thinking about it, I think I...After several days of thinking about it, I think I have a way of doing it . It involves using an extra card that is not an ace or a king (a joker, for instance), and then using inclusion-exclusion. Now I have to actually do the grunge to see if it works out... <br /><br />Goulden's solution is a thing of beauty!jasonhttp://www.blogger.com/profile/15972096640733349029noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-91149751001707063782010-01-20T16:39:34.940-05:002010-01-20T16:39:34.940-05:00I have always liked problems about probability. Ho...I have always liked problems about probability. However, after facing, a short time ago, "The Monty Hall problem", and failing miserably in it (after, obviously, have refused to understand and accept its right solution for a while), I'm in a sort of a temporary (I hope) lack-of-confidence crisis.Filipe Calvario (from Brazil)noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-58055115358199627162010-01-20T06:35:24.625-05:002010-01-20T06:35:24.625-05:00Student:
Here it's not so much a matter of an...Student:<br /><br />Here it's not so much a matter of answering "correctly" - even I had an outline of an approach that would produce the correct answer, but the details would have been messy. It's more a matter of finding the combinatorially-satisfying solution. You also have to consider how much time you are willing to spend on it. <br /><br />Ian Goulden, since he's an expert, could come up with the combinatorially-satisfying solution quickly. My guess is that many mathematicians could do it, but it might take a <i>lot</i> longer for them.Jeffrey Shallithttp://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-59699964253422761952010-01-19T23:27:45.759-05:002010-01-19T23:27:45.759-05:00i'm a bit curious, if you had instead asked a ...i'm a bit curious, if you had instead asked a faculty from our from our actsci & statistics department, would he/she been able to answer correctly?studentnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-63048085738174760272010-01-19T17:42:40.917-05:002010-01-19T17:42:40.917-05:00You know you're at a great school when it has ...You know you're at a great school when it has a whole department dedicated to combinatorics and optimization. Combinatorics has a never-ending supply of problems, like the one above, that are guaranteed to keep you up at night and make you neglect your family.R0bnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-32330026735717603792010-01-19T17:28:08.477-05:002010-01-19T17:28:08.477-05:00I was about to demonstrate how a Monte Carlo metho...I was about to demonstrate how a Monte Carlo method would achieve the same result especially as you had the final value approximated to 4 decimals (for some reason I often see 6 for such values).<br /><br />However, while obtaining a decent amount of certainty for 3 digits is almost instantaneous, getting 4 digits takes on the order of 10 minutes. Which is no fun.<br /><br />The Monte Carlo method gives no deeper level of understanding (as with the generalized equation you presented), it has helped me in the past win over "Monty-Hall-deniers" <br /><br />And it has confirm your 0.4863D. Swartnoreply@blogger.com