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 Tags math puzzles , probability puzzles

 28th January 2003, 09:04 PM #1 Skeptoid Graduate Poster   Join Date: Sep 2002 Posts: 1,532 Football pool probability question I originally posted this question in a Banter Superbowl thread that quickly got buried on page 2. Perhaps it'll have a better chance of being seen by someone who can do probabilities here. I was in a standard 100 square pool with the following variation: While our positions in the grid remained constant we were issued 4 sets of scores, one set for each quarter. What is the probability that I would receive an identical set of scores for two quarters? My intuition tells me that the probability is rather small and I'm suspicious of hanky-panky. Are my suspicions warranted?
 28th January 2003, 10:48 PM #2 Walter Wayne Wayne's Words     Join Date: Jul 2002 Location: Ottawa, ON Posts: 2,439 Chance of not having any duplicates given a hundred squares. 94.1% (1.00*.99*.98*.97) Odds of of having at least one duplicate 5.9% or 1 in 17 (1 - 0.941) What is the probability that I would receive an identical set of scores for two quarters? 5.8% (4 choose 2) * 100 * 99 * 98 / (1004) In the second case I am disregarding the odds of getting 3 similar or 4 similar.
 28th January 2003, 11:57 PM #3 Skeptoid Graduate Poster   Join Date: Sep 2002 Posts: 1,532 Thanks, Walt. Suppose, for the sake of discussion and to be more general, that I were in n distinct 100 square pools (the original problem being n = 4). Further suppose that n is even. What is the probability that half of my scores would be identical? Would it be the same (5.8%)?
 29th January 2003, 08:11 AM #4 Ladewig Hipster alien     Join Date: Dec 2001 Location: not measurable Posts: 16,789 Quote: I'm suspicious of hanky-panky. Always consider misfeasance before malfeasance. The world has more ignorance than evil.
 29th January 2003, 02:41 PM #5 Walter Wayne Wayne's Words     Join Date: Jul 2002 Location: Ottawa, ON Posts: 2,439 Hi Skeptoid I won't have a chance to answer this before heading to the Amazing meeting. If it isn't answered when I get back, I will write down a general solution for any given n. However the answer for n = 6 and 3 identical is 100*(6 choose 3)*99*98*97/1006 This equal 0.19 % Didn't have a chance to check my math, but I will get back to you. Walt edited to add: if you are using this because you think someone fixed something, wait until someone else has checked this or I have a chance to check my math.
 29th January 2003, 05:38 PM #6 Skeptoid Graduate Poster   Join Date: Sep 2002 Posts: 1,532 Thanks again, Walt. In the Superbowl thread, Diezel said he's seen 2 of 4 the same before so, if your math is correct, then my suspicions of a fix are not really warranted. I was thinking that the probability was probably less than 1%. Not being a regular at the establishment where the pool was taken up, my eyebrow raised when I got a pair of really lousy scores (2 - 5). I'm not normally a suspicious person but when large amounts money are involved ... A good lesson in critical thinking though. I was skeptical of my own intuition and sought answers for or against before I acted. If you want to post the more general solution it would be interesting but I can see that as n gets large, the probability approaches 0.
 6th February 2003, 07:23 PM #7 Walter Wayne Wayne's Words     Join Date: Jul 2002 Location: Ottawa, ON Posts: 2,439 Because I said I would ... Looking at the above equations that I posted, I believe they are wrong. I believe they slightly over estimate the probability as they count some combinations twice. To simplify the equation I will make sure I over estimate and just calculate an upper bound on the probability. p(1/2 are the same) > 100*(N choose N/2)*99N/2/100N Walt
 7th February 2003, 08:25 PM #8 Number Six JREF Kid     Join Date: Sep 2001 Posts: 5,017 Are you talking about buying one square in n distinct pools, each of which gives you a different set of numbers for each of the four quarters? Or are you talking about buying n squares in one pool that gives you a different set of numbers for each of the four quarters? Or are you talking about something else? Figuring out exactly what the question is seems simple but a lot of the time it is harder than it seems.
 11th February 2004, 11:11 AM #9 xouper Banned   Join Date: Jan 2002 Posts: 2,189 bump

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