Two Plus Two Newer Archives probability of n ties in a race to three (roshambo)
 FAQ Members List Calendar Search Today's Posts Mark Forums Read

#1
11-29-2007, 02:56 AM
 imfatandugly Senior Member Join Date: Jul 2005 Posts: 267
probability of n ties in a race to three (roshambo)

Too lazy to figure this out. I was playing rock, paper scissors with a friend and had an improbable amount of ties. Given that two people randomly pick paper, rock, scissors what is the probability that there will be n ties before someone wins twice? How about n or less?

thanks.
#2
11-29-2007, 03:35 AM
 pzhon Senior Member Join Date: Mar 2004 Posts: 4,515
Re: probability of n ties in a race to three (roshambo)

The number of ties before a decisive throw follows a geometric distribution with mean 1/2. There is a 50% chance that a best-of-three-decisive-throws match will last 2 decisive throws, and a 50% chance that it lasts 3. So, the distribution is 50% of a convolution of 2 geometric distributions of mean 1/2 plus 50% of a convolution of 3 geometric distributions of mean 1/2.

P(n) = 1/2 (n+1 C 1) (1/3)^n (2/3)^(2) + 1/2 (n+2 C 2) (1/3)^n (2/3)^3.
#3
11-29-2007, 09:24 PM
 Siegmund Senior Member Join Date: Feb 2005 Posts: 1,850
Re: probability of n ties in a race to three (roshambo)

You can replace the phrase "convolution of n geometric distributions" with "negative binomial distribution" (and "with mean 1/2" by "with probability of success 2/3") to make it look a little less scary.

But it's the same calculation.
#4
11-29-2007, 11:42 PM
 imfatandugly Senior Member Join Date: Jul 2005 Posts: 267
Re: probability of n ties in a race to three (roshambo)

thanks, sometimes I think phzon forgets that he isn't talking to grad students. Is there an easy way to see this? Link to derivation...

phzon...when you look at problems like this are you like "duh.... so obviously geometric distribution, and so obvious that it is equally like to end in two decisive throws as three."

Thanks though, your input is always appreciated.
#5
11-30-2007, 11:37 AM
 rufus Senior Member Join Date: Aug 2005 Posts: 425
Re: probability of n ties in a race to three (roshambo)

[ QUOTE ]
thanks, sometimes I think phzon forgets that he isn't talking to grad students. Is there an easy way to see this? Link to derivation...

phzon...when you look at problems like this are you like "duh.... so obviously geometric distribution, and so obvious that it is equally like to end in two decisive throws as three."

[/ QUOTE ]

You can think of this sort of question in terms of transitions. On a throw-by-throw level, each result (p1 wins, p2 wins, tie) has a chance of 1/3 of occuring, so you could work it out like a 3-dimensional pascal's triangle.
#6
11-30-2007, 09:46 PM
 pzhon Senior Member Join Date: Mar 2004 Posts: 4,515
Re: probability of n ties in a race to three (roshambo)

[ QUOTE ]
thanks, sometimes I think phzon forgets that he isn't talking to grad students.

[/ QUOTE ]
I hope it doesn't take a graduate student to evaluate (n+2) choose 2, the most complicated part of the formula I gave.

When I give an answer, I often expect an interested reader to take some time to think about what I said. The amount of time it takes depends on the reader's interest and background. If you expect to get everything at the first reading, you will be disappointed in mathematics.

Your complaints made me regret helping you again. I'm not inclined to spoonfeed you, and if I were, what I would need to say to make everything clear to you would depend on your background, which I don't know.

[ QUOTE ]
when you look at problems like this are you like "duh.... so obviously geometric distribution, and so obvious that it is equally like to end in two decisive throws as three."

[/ QUOTE ]
No. Here is a rough transcript of my thought processes: "Is there anything ifu could have meant other than best of 3 decisive results? I don't think so. After 2 decisions, there is a 50% chance it's over, and a 50% chance it will take one more. The number of ties before each decisive result is geometric. What is the convolution of geometric distributions called again? "Hypergeometric" would make sense, but I think that's something else. Yes, Wikipedia says hypergeometric is something else. Oh well. It's easy to read off the explicit formula from basic combinatorics, but that's prone to producing errors. Let me put the formula into Mathematica to check that the total probability is 1. Yup. Check a few values for plausibility. Hmm, the denominators for P(0) and P(1) are equal to 27, but that pattern can't continue. The expected value is 5/4. That makes sense, as that is the average of 2/2 and 3/2. Post."
#7
11-30-2007, 10:20 PM
 imfatandugly Senior Member Join Date: Jul 2005 Posts: 267
Re: probability of n ties in a race to three (roshambo)

[ QUOTE ]
[ QUOTE ]
thanks, sometimes I think phzon forgets that he isn't talking to grad students.

[/ QUOTE ]
I hope it doesn't take a graduate student to evaluate (n+2) choose 2, the most complicated part of the formula I gave.

When I give an answer, I often expect an interested reader to take some time to think about what I said. The amount of time it takes depends on the reader's interest and background. If you expect to get everything at the first reading, you will be disappointed in mathematics.

Your complaints made me regret helping you again. I'm not inclined to spoonfeed you, and if I were, what I would need to say to make everything clear to you would depend on your background, which I don't know.

[ QUOTE ]
when you look at problems like this are you like "duh.... so obviously geometric distribution, and so obvious that it is equally like to end in two decisive throws as three."

[/ QUOTE ]
No. Here is a rough transcript of my thought processes: "Is there anything ifu could have meant other than best of 3 decisive results? I don't think so. After 2 decisions, there is a 50% chance it's over, and a 50% chance it will take one more. The number of ties before each decisive result is geometric. What is the convolution of geometric distributions called again? "Hypergeometric" would make sense, but I think that's something else. Yes, Wikipedia says hypergeometric is something else. Oh well. It's easy to read off the explicit formula from basic combinatorics, but that's prone to producing errors. Let me put the formula into Mathematica to check that the total probability is 1. Yup. Check a few values for plausibility. Hmm, the denominators for P(0) and P(1) are equal to 27, but that pattern can't continue. The expected value is 5/4. That makes sense, as that is the average of 2/2 and 3/2. Post."

[/ QUOTE ]

Obviously it wasn't the binomial that was giving trouble. It was the "distribution is 50% of a convolution of 2 geometric distributions ". Yeah I looked it up and I get what it means.
Your help is always appreciated, like i've said a dozen times. Just sometimes your answers seem to come out of thin air, which is frustrating because I don't want to just know the answer, I want to know why it is the answer. Like you said it's hard to make things clearer if you don't the background of the person your responding to.
I should of thought of that. I guess i'm just used to answers with proofs or lines of reasoning (makes the answers seem like they haven't fallen out of the sky). I'm sure when you talk to whoever in real life if you spout out an answer and they ask you to explain your reasoning you don't get offended.....

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off Forum Rules
 Forum Jump User Control Panel Private Messages Subscriptions Who's Online Search Forums Forums Home Two Plus Two     Two Plus Two Internet Magazine     The Two Plus Two Bonus Program     Special Sklansky Forum     About the Forums     MOD DISCUSSION     Test General Poker Discussion     Beginners Questions     Books and Publications     Televised Poker     News, Views, and Gossip     Brick and Mortar     Home Poker     Poker Beats, Brags, and Variance     Poker Theory     Poker Legislation Coaching/Training     Stoxpoker.com     DeucesCracked.com German Forums     Poker Allgemein: Poker in general     Strategie: Holdem NL cash [German]     Strategie: Sonstige     Internet/Online [German]     BBV [German]     Small Talk [German] French Forums     Forum Francophone     Strategie [French]     BBV [French] Limit Texas Hold'em     Texas Hold'em     High Stakes Limit     Medium Stakes Limit     Small Stakes Limit     Micro Stakes Limit     Mid-High Stakes Shorthanded     Small Stakes Shorthanded     Limit-->NL PL/NL Texas Hold'em     High Stakes     Medium Stakes     Small Stakes     Micro Stakes     Full Ring Tournament Poker     MTT Strategy     High Stakes MTT     MTT Community     STT Strategy     Tournament Circuit/WSOP Other Poker     Omaha/8     Omaha High     Stud     Heads Up Poker     Other Poker Games General Gambling     Probability     Psychology     Sports Betting     Other Gambling Games     Entertainment Betting     Money Making and Other Business Discussion Internet Gambling     Internet Gambling     Internet Bonuses     Affiliates/RakeBack     Software     Poker Site Software, Skins, & Networks 2+2 Communities     Other Other Topics     The Lounge: Discussion+Review     EDF     BBV4Life Other Topics     Sporting Events     Politics     Business, Finance, and Investing     Travel     Science, Math, and Philosophy     Health and Fitness     Student Life     Golf     Video Games     Puzzles and Other Games     Laughs or Links!     Computer Technical Help     Bin Sponsored Support Forums     RakebackNetwork     RakeBackDepot     RakeReduction.com Rakeback     PokerSavvy

All times are GMT -4. The time now is 02:03 PM.