Two Plus Two Poker Forums How do you determine if a coin is fair?
 User Name Remember Me? Password
 Register FAQ Search Today's Posts Mark Forums Read Video Directory TwoPlusTwo.com

 Notices

 Probability Discussions of probability theory

 Thread Tools Display Modes
 01-06-2012, 03:35 PM #1 Riorin journeyman   Join Date: Nov 2005 Posts: 272 How do you determine if a coin is fair? Let's say someone claims the coin is fair - 50% heads, 50% tails? You toss it: a) 3 times - 3 heads b) 9 times - 9 heads c) 100 times - 100 heads I guess getting 3 heads out of 3 tosses wouldn't surprise anyone, getting 9 heads out of 9 tosses would raise an eyebrow, getting 100 heads out of 100 tosses from presumably fair coin would certainly be tagged as cheating. At what number/point would you determine the coin not to be fair? Or to paraphrase a question (superturbos run mostly on all-in EV luck) at what relation of your -EV running vs. stakes could you/would you conclude that such amount of bad luck is "very, very, very.... improbable" thus cheating?
 01-06-2012, 08:09 PM #2 statmanhal Pooh-Bah   Join Date: Jan 2009 Posts: 3,964 Re: How do you determine if a coin is fair? There is a statistical field called Inference and one of the main parts is Testing Hypothesis. For the fair coin question, the hypothesis is that the coin is fair, or equivalently, the probability the coin will fall heads is 0.50. An experiment is run, namely tossing the coin a number of times, and the hypothesis is tested by comparing the results to that which would be expected if the hypothesis were true. Since the experiment involves a sample that produces random results, one can rarely be certain that the results absolutely confirm or deny the validity of the hypothesis, so statisticians use such concepts as confidence and significance levels to validate or refute the hypothesis. A simple test for the fair coin, is to toss it n times. Assume r successes (heads) were observed. Then the estimated probability of a head is r/n. The following t statistic is then computed: t = (r/n -0.5) / [sqrt(0.5^2/n] The numerator is the difference between the observed and hypothesized probability of a head. The denominator is the standard deviation of the proportion of heads if the hypothesis was true. This t statistic is compared to a critical t value based on the desired significance level and sample size to determine if the observed proportion of heads is significantly different from the hypothesized value of 0.50. Note: The above uses the normal approximation to the binomial distribution. There are other tests that are more sophisticated, but the above is usually satisfactory if the sample size, n, is not too small, say over 30. Another approach is to use the Excel binomial distribution function. If r heads are observed in n trials, enter the following in a cell: Binomdist(r, n, 0.5, 1) This is the probability that r or fewer heads would be observed in n tosses if the true probability of a head was 0.5. If this probability is less than your significance level, e.g., 5%, then you reject the hypothesis that the coin is fair. Last edited by statmanhal; 01-06-2012 at 08:21 PM.
 01-06-2012, 11:30 PM #3 NewOldGuy Pooh-Bah   Join Date: Mar 2009 Location: In the wires Posts: 5,014 Re: How do you determine if a coin is fair? I think just knowing the probability of the streak happening isn't enough. We also need to have some idea of the trial universe we are working in. For example, 1 in a billion events have certainly happened multiple times on Poker Stars, but probably not on NakedPoker. If one happened twice on NakedPoker it would be suspicious (but once really means nothing). Likewise, since 9 heads in a row happens every 512 flips on average, if we see this happen once it means nothing, but if we see it happen multiple times in a small number of trials, or the streak keeps getting longer, it would be noteworthy. And 100 in a row probably hasn't happened in the history of the universe in any kind of 50/50 wager, coins or whatever. Simply because there haven't been enough trials. If something has a one in a googillion chance, then on average that is how many trials it takes for the first occurrence.
 01-07-2012, 11:22 AM #4 AaronBrown Pooh-Bah     Join Date: May 2005 Location: New York Posts: 4,240 Re: How do you determine if a coin is fair? To fully specify the problem you need to know two more things: the prior probability that the coin is fair (this is was NewOldGuy is referring to) and the gains and losses from decisions. For example, suppose you have a gamble where you pay \$10 if you lose a coin flip and win \$11 if you win. You think there's a 98% chance the game is honest, but a 2% chance that the coin is weighted so you always lose. Your EV per flip is +\$0.50 in an honest game, -\$10 in a dishonest one. Since there's a 98% chance the game is honest, your EV for the first flip is 0.98*\$0.50 - 0.02*\$10 = +\$0.29. If you lose the first flips, the chance that the game is rigged increases to 3.92% (out of 100 games, 2 will be rigged and 98 fair, in 49 of the 98 fair games you win the first flip, so if you lose the remaining chance is 2 / 51 = 3.92% that the game is rigged). So your EV of a second flip given that you lose the first one is 0.9608*\$0.50 - 0.0392*\$10 = \$0.09. If you lose the first two flips, the game becomes negative EV. You're not sure the coin is rigged, 25% of the time you lose the first two flips in an honest game. But your suspicion level has increased enough that it no longer makes sense to play.
 01-07-2012, 09:15 PM #5 Sherman Carpal \'Tunnel     Join Date: Jun 2005 Location: Psychology Department Posts: 7,764 Re: How do you determine if a coin is fair? This is probably a good thread to ask this question. Let's say we know that in the population of coins any given coin can have a "true" heads value of 0% to 100%, but that the expected value (mean) is 50% and the coins are normally distributed. Question 1: What is the variance (or SD) of this population? How do you compute it? Question 2: What is the expected variance (or SD) for a coin drawn from this population after three flips? Now let's say we have a coin from this population and we witness it land on heads three times. Question 3: Given the answers to questions 1 and 2, can we now combine these two pieces of information to make a prediction about the next coin flip (as follows): (popMean / popVar + obsMean / expVar) / (1/popVar + 1/expVar) = p(Next flip heads) And: 1 / (1/popVar + 1/expVar) = Variance of p(Next flip heads) Where popMean is the known population mean (in this case .50), popVar is the known population variance (Question 1), obsMean is the observed mean (in this case 1.0), and expVar is the expected variance for the observed mean (Question 2). If this is not right, can someone please explain what is wrong and/or how it can be fixed.
01-07-2012, 09:19 PM   #6
NewOldGuy
Pooh-Bah

Join Date: Mar 2009
Location: In the wires
Posts: 5,014
Re: How do you determine if a coin is fair?

Quote:
 Originally Posted by Sherman Let's say we know that in the population of coins any given coin can have a "true" heads value of 0% to 100%, but that the expected value (mean) is 50% and the coins are normally distributed.
That makes the coin population as a whole equivalent to a random number generator.

01-07-2012, 09:22 PM   #7
Sherman
Carpal \'Tunnel

Join Date: Jun 2005
Location: Psychology Department
Posts: 7,764
Re: How do you determine if a coin is fair?

Quote:
 Originally Posted by NewOldGuy That makes the coin population as a whole equivalent to a random number generator.
Yes. With a mean of .50.

01-07-2012, 11:04 PM   #8
RustyBrooks
Carpal \'Tunnel

Join Date: Feb 2006
Location: Austin, TX
Posts: 23,122
Re: How do you determine if a coin is fair?

Quote:
 Originally Posted by Sherman Yes. With a mean of .50.
This should make it clear why you can't "calculate" the variance, because it's not a fixed emergent property of the population of coins you describe. You could pick any variance you wanted and construct a population of coins that fits your description. It's a little like saying "I have a population of people with heights ranging from 3 to 8 feet, with an average of 5'10". What's the variance?"

01-07-2012, 11:07 PM   #9
Sherman
Carpal \'Tunnel

Join Date: Jun 2005
Location: Psychology Department
Posts: 7,764
Re: How do you determine if a coin is fair?

Quote:
 Originally Posted by RustyBrooks This should make it clear why you can't "calculate" the variance, because it's not a fixed emergent property of the population of coins you describe. You could pick any variance you wanted and construct a population of coins that fits your description. It's a little like saying "I have a population of people with heights ranging from 3 to 8 feet, with an average of 5'10". What's the variance?"
Well it can't be any number. There is a limit of 0% to 100% heads.

But I understand. This was the problem I ran into as well.

So let's change the problem a bit and make another assumption. Let's say .5% of coins land on tails 100% of the time and .5% land on heads 100% of the time. The other 99% of coins are exactly fair. What's the variance here? And if we can solve for it (and assume our coin comes from this population) does the rest of my example hold?

 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 Links to Popular Forums     News, Views, and Gossip     Beginners Questions     Marketplace & Staking     Casino & Cardroom Poker     Internet Poker     NL Strategy Forums     Poker Goals & Challenges     Las Vegas Lifestyle     Sporting Events     Politics     Other Other Topics Two Plus Two     About the Forums     Two Plus Two Magazine Forum     The Two Plus Two Bonus Program     Two Plus Two Pokercast     The Best of Two Plus Two Marketplace & Staking     Commercial Marketplace     General Marketplace     Staking - Offering Stakes     Staking         Staking - Offering Stakes         Staking - Seeking Stakes         Staking - Selling Shares - Online         Staking - Selling Shares - Live         Staking Rails         Transaction Feedback & Disputes     Transaction Feedback & Disputes Coaching & Training     Coaching Advice     Cash Game Poker Coach Listings     Tournament/SNG Poker Coach Listings Poker News & Discussion     News, Views, and Gossip     Poker Goals & Challenges     Poker Beats, Brags, and Variance     That's What She Said!     Poker Legislation & PPA Discussion hosted by Rich Muny     Twitch - Watch and Discuss Live Online Poker     Televised Poker     Two Plus Two Videos General Poker Strategy     Beginners Questions     Books and Publications     Poker Tells/Behavior, hosted by: Zachary Elwood     Poker Theory     Psychology No Limit Hold'em Strategy     Medium-High Stakes PL/NL     Micro-Small Stakes PL/NL     Medium-High Stakes Full Ring     Micro-Small Stakes Full Ring     Heads Up NL     Live Low-stakes NL Limit Texas Hold'em Strategy     Mid-High Stakes Limit     Micro-Small Stakes Limit Tournament Poker Strategy     STT Strategy     Heads Up SNG and Spin and Gos     Mid-High Stakes MTT     Small Stakes MTT     MTT Community Other Poker Strategy     High Stakes PL Omaha     Small Stakes PL Omaha     Omaha/8     Stud     Draw and Other Poker Live Poker     Casino & Cardroom Poker         Venues & Communities         Regional Communities     Venues & Communities     Tournament Events         WPT.com     Home Poker     Cash Strategy     Tournament Strategy Internet Poker     Internet Poker         Winning Poker Network         nj.partypoker.com         Global Poker     Commercial Software     Software         Commercial Software         Free Software General Gambling     Backgammon Forum hosted by Bill Robertie.     Probability     Sports Betting     Other Gambling Games 2+2 Communities     Other Other Topics         OOTV         Game of Thrones     The Lounge: Discussion+Review     EDF     Las Vegas Lifestyle     BBV4Life         omg omg omg     House of Blogs Sports and Games     Sporting Events         Single-Team Season Threads         Fantasy Sports     Fantasy Sports         Sporting Events     Wrestling     Golf     Chess and Other Board Games     Video Games         League of Legends         Hearthstone     Puzzles and Other Games Other Topics     Politics     History     Business, Finance, and Investing     Science, Math, and Philosophy     Religion, God, and Theology     Travel     Health and Fitness     Laughs or Links!     Computer Technical Help     Programming International Forums     Deutsch         BBV [German]     Français     Two Plus Two en Espańol

All times are GMT -4. The time now is 07:25 AM.

 Contact Us - Two Plus Two Publishing LLC - Privacy Statement - Top