Author: Tom Kerrigan
Date: 20:50:49 07/25/03
Go up one level in this thread
On July 25, 2003 at 22:31:04, Eric Campos wrote: >On July 25, 2003 at 14:31:58, Tom Kerrigan wrote: > >>On July 25, 2003 at 07:14:19, George Tsavdaris wrote: >> >>>On July 24, 2003 at 23:46:50, Tom Kerrigan wrote: >>> >>>>Given a round robin tournament with 13 participants of equal strength, 5 of >>>>which are The King: >>>> >>>>There's a 14% chance The King would take the top 3 spots. >>> >>>I don't know for sure***, but i think you are wrong. >>> >>>Because if the participants have equal strength we can do the following: >>> >>>If N is the number of all possible classifications of the programs >>>then N = 13!/5! as we have five "The King". >> >>As Ricardo Gibert pointed out, this doesn't take into account the possibility of >>draws. I imagine the program I wrote is reasonably accurate. >> >>-Tom > >When talking about the probability of a certain program coming in 1st or 2nd >place, I don't think the possibility of draws comes into play. I understand >that the possibility of "ties" in the ranking does come into play, but that's a >different story. I'm sorry, I meant the possibility of ties. Saying draws is ambiguous. >I have to agree with George's math. As a check, the prob(1st and 2nd and 3rd) >is 5/13 * 4/12 * 3/11, which gives the same result of 3.50%. This ignores the >fact that programs can tie in the rankings, e.g. CM could tie another program >for 3rd / 4th place, but we haven't defined how to treat this case anyway. If CM ties for 3rd place, it still gets 3rd place. >When I check your 11.4% expected 1st place outcome for other programs, something >doesn't add up. 11.4% * 8 = 91.2%, which leaves CM only an 8.8% chance of >clinching 1st. The probability of getting 1st, 2nd and 3rd must be less than >this, but your simulation shows 14%. Perhaps I just misunderstand what you >meant with your 11.4% number. You're ignoring the possibility of ties. When you allow ties, the number of places that the programs compete for (including 1st place) effectively decreases, so it's no longer 1/13, but 1/(<13). Interestingly, the 11.4% indicated by my program is exactly 1.5*7.6%. This is probably for another obvious reason that I'm overlooking. >The probability of a non-King program winning is still 1/13 (7.69%). No, seems like it should be > 50% at least. >Consider ties: >-------------- >The probability of sole possession of 1st place is lower than 7.69%. >The probability of 1st place or sharing 1st place is greater than 7.69%. >So the answer depends on how you want to define 1st place. > >One last thought: If all of the programs had equal capability, the odds of The >King getting 1st, 2nd, and 3rd is exactly equal to the odds of it capturing the >last three places! Sure. This would be easy to test with my program but right now I'm using a different computer. I can run the test later. -Tom
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