Probability and Card Shuffling
Date: Thu, 3 Nov 1994 12:30:41 +0100 From: Henrik Johansson Subject: Probability question Intro: if I have n shuffled cards, numbered from 0 to n-1, the probability that no card will have the same number on it as its position in the deck is not very hard to figure out... Problem: If I have n shuffled cards in k colours numbered from 0 to (n/k)-1, what is the probability that no card will have the same number on it as its position in the deck modulo n/k? Standard example: n = 52, k = 4, cards numbered from 0 to 12. A card with number 3 mustn't be on positions 3, 16, 29, or 42. I stumbled onto this problem a few years ago but haven't managed to solve it. Neither has any of my teachers. A simulation gave that about one shuffle out of 64 (if I remember correctly) was successful. /hj
Date: Thu, 3 Nov 1994 08:51:31 +0900 X-Sender: email@example.com Hi Henrik- Thanks for writing to us. I played with this problem for a little while, but I didn't get very far. One way to look at it is to split up the shuffled deck into n/k piles. This makes the problem look a little more like the "Intro" problem that you gave - remember - the intro problem is just the same as the harder one when k=1. There are more doctors coming on call all day. Maybe one of them can help. Thanks again for writing to us, Margaret M.D. on call
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