Do you believe in Luck?

[mutile]

That is how randomness works, I wouldn't call it an ability, just the definition of what randomness is.

 

[Complexity]

Possibly, but we certainly can't deny that some people have an uncanny ability to have the outcome go in their favor mnore often than others.

Yes, we can. Things happen. No one has 'an uncanny ability...'

Don't be ridiculous.

 

[Beth]

More Data

Okay, it's been a while. Let me bring you up-to-date and see if any one has an explanation for these results other than random chance.

Hubby hasn't played much poker of late due to many other things going on in our lives. However, he started recording all all-in's rather than just the ones we refer to as 'races'. If you don't recall, that was defined as a pair against two over cards.

He's had 10 all-in's and lost nine of them. Here are the hands:

In September, he had one all-in from an on-line game. He went all in after the flop with AK. His opponent had 3, 7. We estimate the probability of loss at 1/3. The flop came 3, 8, 7. He lost.

In October, he played twice with his buddies and had three all-in's each night.

First set of games, at PB, he went all-in

10, 10 against 5, 8. We estimate the probability of loss at 1/3. He lost.
K, K against Q, Q. We estimate the probability of loss at 1/5. He lost.
K, T against A, T. We estimate the probability of loss at 2/3*. He lost.

Second set of games, at BG, he went all-in

A, A against 9,9. We estimate the probability of loss at 1/5. He lost.
A, T against A, K. We estimate the probability of loss at 2/3. He lost.

The following hand is the only non-pre-flop all-in. In this case, he had
A, 6 against K, 3. They went all-in after the turn with 6,3,6,3 showing. We estimate the probability of loss at 0.02. He lost. The river was a 3.

November, he's had two poker nights.

First set of games, at PB, he went all-in
J, J against Q,9. We estimate the probability of loss at 1/5. He won this hand!
A, K (suited) against Q, Q**. We estimate the probability of loss at 1/2. He lost.

Second set of games, at BG, he went all-in only once.
A, A against 9,9***. We estimate the probability of loss at 1/5. He lost.

My computation of the probability of getting one win out of those ten games as 0.00003.

This was computed as 10 * 1/3 * 1/5 * 1/3 * 2/3 * 1/5 * 1/3 * 1/50 * 4/5 * 1/2 * 1/5.

So, any ideas? Are those probabilities reasonable?


* After the loss of KK against QQ, he only had a few blinds left and thought K, T was as good a hand to push with as he was likely to get.

** This qualifies as a race. Our statistics are now 22 wins out of 57 races which has a p-value of .0556.

***Incidentally, and presumably coincidentally, the person with the pocket nines was the same opponent who beat his pocket aces with pocket nines at BG's in October.

 

[Cainkane1]

Its hard not to believe in luck but overall theres no such thing.

 

[Antiquehunter]

Okay, it's been a while. Let me bring you up-to-date and see if any one has an explanation for these results other than random chance.

Hubby hasn't played much poker of late due to many other things going on in our lives. However, he started recording all all-in's rather than just the ones we refer to as 'races'. If you don't recall, that was defined as a pair against two over cards.

He's had 10 all-in's and lost nine of them. Here are the hands:

In September, he had one all-in from an on-line game. He went all in after the flop with AK. His opponent had 3, 7. We estimate the probability of loss at 1/3. The flop came 3, 8, 7. He lost.

If he went all in AFTER the flop, he was WAY behind. He was dead to perfect cards, unless he had a backdoor flush opportunity. Off the top of my head without checking a calculator, he had about a 18% chance on the turn of even being alive (must catch an A, 8 or a K), and then about a 16% chance to catch on the river assuming he wasn't drawing dead on the turn. Combined, this is MAYBE 20-22% pot equity. Less than 1/3 and not a very desirable position to be in.

In October, he played twice with his buddies and had three all-in's each night.

First set of games, at PB, he went all-in

10, 10 against 5, 8. We estimate the probability of loss at 1/3. He lost.
K, K against Q, Q. We estimate the probability of loss at 1/5. He lost.
K, T against A, T. We estimate the probability of loss at 2/3*. He lost.

Pair over two undercards (assuming they're suited) you're 80:20 favorite (approx - can be some fluctuation, you need to use a poker hand calculator for more accurate)

Higher pair vs lower pair you are 80:20 - so here you're about right.

Domination in the example at #3 is about 66:33 - so here you're about right.

All the above assume you have gone all-in pre-flop, in 'cleanly' dealt hands, without any shenanigans.

A, A against 9,9. We estimate the probability of loss at 1/5. He lost.
A, T against A, K. We estimate the probability of loss at 2/3. He lost.

Your AT vs AK the odds are probably closer to around 70:30 against you, but good enough for government work

The following hand is the only non-pre-flop all-in. In this case, he had
A, 6 against K, 3. They went all-in after the turn with 6,3,6,3 showing. We estimate the probability of loss at 0.02. He lost. The river was a 3.

Yup - guy caught a 1-outer. It happens. It was a 1:46 shot.

November, he's had two poker nights.

First set of games, at PB, he went all-in
J, J against Q,9. We estimate the probability of loss at 1/5. He won this hand!
A, K (suited) against Q, Q**. We estimate the probability of loss at 1/2. He lost.

In the first example he was a 70:30 favorite. 2nd example is a coin-toss.

Second set of games, at BG, he went all-in only once.
A, A against 9,9***. We estimate the probability of loss at 1/5. He lost.

My computation of the probability of getting one win out of those ten games as 0.00003.

This was computed as 10 * 1/3 * 1/5 * 1/3 * 2/3 * 1/5 * 1/3 * 1/50 * 4/5 * 1/2 * 1/5.

So, any ideas? Are those probabilities reasonable?

Well, I would interpret this run of cards as a little bit unlucky I suppose, but certainly not indicative of some sort of cloud of poor luck that is following him around.

Again, I assume here you've been assiduous with data collection - the scenario you describe at #1 makes a big difference (going all-in post-flop) as you can see. You can't measure that hand as AK vs 73, once the 73 has flopped two pair.

A couple of general observations that may get better results:

- in the AK vs 73 example, did he make a significant raise pre-flop to try and drive out crap hands? If not, this is how one gets into trouble, by allowing junk to beat you cheaply.

- You said he just sort of chucked in his last few blinds (and was an underdog) - I don't know what sort of stakes he's playing at, but this is a 'loser' move. If he has lost for the evening (at a cash game) and isn't motivated to keep playing, he should pickup his chips and save them for next time / cash in whatever he has, rather than throw them away or shove out of desperation. In tournaments, one is forced to do this, but you need to look at cash games as a continuous run of never-ending games - and its important to be properly funded each time you sit down, as stack-size matters in poker, big-time.

- In general he seems to be making not bad pre-flop decisions and generally getting involved in preflop contests where he is a reasonable favorite. Only one 'race' and only 3 mistakes. So, in 60% of all-in decisions, he's making 'correct' decisions. That's not a bad start.

Another way to look at this - lets say your state-run scratch & win lotto advertises 1:8 tickets is a winner (of some prize). If you bought 8 tickets, you could reasonably expect to win on at least one of those tickets. In this case, maybe he's bought 10 and hasn't won yet. He's been a little unlucky over an isolated set of examples.

I would expect that if he is making decisions such as described here, the results will revert to the norm, over time. And just as its rather rare to spot a run of 10 hands where he's only won one, play enough of these sequences, and he'll have one where he wins 9.

You mentioned at one point that the same guy has beaten him with an underpair (which happened to be 9's) twice now. This is hardly a pattern to be excited about, but I would make sure you're following good anti-card cheating mechanisms even at a home game. Use a cut card to make sure the deck is cut before each deal, and keep the cut card on the bottom of the deck. This prevents a lot of the more common ways people try to cheat with cards - and yes, unfortunately, sometimes people DO cheat even at 'the friendly' game for relatively low stakes. It can be like a disease.

/snip

 

[Beth]

Thanks Antiquehunter. I was hoping you would reply with an evaluation of the odds we were assuming.

If he went all in AFTER the flop, he was WAY behind.

Sorry, that was my error in typing it up. I wrote after, but should have said before. This was a pre-flop all-in, not post flop. Is 1/3 a fair assessment of that probability?

Pair over two undercards (assuming they're suited) you're 80:20 favorite (approx - can be some fluctuation, you need to use a poker hand calculator for more accurate)

All the above assume you have gone all-in pre-flop, in 'cleanly' dealt hands, without any shenanigans.

Thanks for the confirmation on the odds. Your assumptions are good. When it wasn't a pre-flop call, I'll specify what cards were already up in order to assess the win/loss probability at the time the all-in was made.

Yup - guy caught a 1-outer. It happens. It was a 1:46 shot.

Yes, it happens. My husband feels such outcomes go against him more often that is reasonable to expect. I have to admit, he's been collecting data on his poker hands for close to a year now and the statistics so far confirm what he's been complaining about for years.

In the first example he was a 70:30 favorite. 2nd example is a coin-toss.

Well, I would interpret this run of cards as a little bit unlucky I suppose, but certainly not indicative of some sort of cloud of poor luck that is following him around.

Recomputing the probability with your suggested changes to the odds:

10 * 1/3 * 1/5 * 1/5 * 2/3 * 1/5 * 2/3 * 1/46 * 7/10 * 1/2 * 1/5 = 0.000018 = 0.0018%

If you don't feel this is indicative of consistently poor luck, let me ask you how you feel that 'poor luck' might be established? What data should be collected and how should it be evaluated? These are serious questions, as that is exactly what we are trying to establish is or isn't happening.

Again, I assume here you've been assiduous with data collection - the scenario you describe at #1 makes a big difference (going all-in post-flop) as you can see. You can't measure that hand as AK vs 73, once the 73 has flopped two pair.

Yes, he's been assiduous with the data collection. Sorry about my error on the description of that first hand.

He gives me the results when he comes home from poker night. He doesn't really play on-line anymore since the U.S. had it's crackdown on on-line poker sites. The on-line tournament in September was all he has in the past few months. That was the point at which he started collecting data on all of his all-in hands. By the way IIRC, this change was due to a suggestion that you made earlier in this thread. Thanks.

A couple of general observations that may get better results:

- You said he just sort of chucked in his last few blinds (and was an underdog) - I don't know what sort of stakes he's playing at, but this is a 'loser' move. If he has lost for the evening (at a cash game) and isn't motivated to keep playing, he should pickup his chips and save them for next time / cash in whatever he has, rather than throw them away or shove out of desperation. In tournaments, one is forced to do this, but you need to look at cash games as a continuous run of never-ending games - and its important to be properly funded each time you sit down, as stack-size matters in poker, big-time.

Yes it was a tournament. That's all he plays with his buddies. They get together twice a month and run two or three tournaments on a Saturday night. Basically, you can assume all the games are tournaments as he isn't currently playing in any other venues.

- In general he seems to be making not bad pre-flop decisions and generally getting involved in preflop contests where he is a reasonable favorite. Only one 'race' and only 3 mistakes. So, in 60% of all-in decisions, he's making 'correct' decisions. That's not a bad start.

Another way to look at this - lets say your state-run scratch & win lotto advertises 1:8 tickets is a winner (of some prize). If you bought 8 tickets, you could reasonably expect to win on at least one of those tickets. In this case, maybe he's bought 10 and hasn't won yet. He's been a little unlucky over an isolated set of examples.

No. This isn't an isolated set of examples. This is ALL of the all-in hands he's had since he decided to record all of them back in September rather than just the 'races'. Please recall that he started collecting data because he felt that his poker outcomes were noticably worse than was reasonble to attribute to random chance. We wanted to test whether his subjective observations have been accurate or whether the results are actually typical of random chance and his feelings could reasonably attributed to confirmation bias, which was what I kept telling him. So far, the data collected is supporting the hypothesis that his observations were accurate and his outcomes are worse than can reasonably be explained by random chance!

I would expect that if he is making decisions such as described here, the results will revert to the norm, over time. And just as its rather rare to spot a run of 10 hands where he's only won one, play enough of these sequences, and he'll have one where he wins 9.

Hopefully soon!

You mentioned at one point that the same guy has beaten him with an underpair (which happened to be 9's) twice now. This is hardly a pattern to be excited about, but I would make sure you're following good anti-card cheating mechanisms even at a home game. Use a cut card to make sure the deck is cut before each deal, and keep the cut card on the bottom of the deck. This prevents a lot of the more common ways people try to cheat with cards - and yes, unfortunately, sometimes people DO cheat even at 'the friendly' game for relatively low stakes. It can be like a disease.

I've occasionally played with them and can verify that they use a cut card and they keep it on the bottom. Interestingly enough, we found out the next day that after he had left (after losing with Ace's to nines), the player who won against him ended up going out the same way with the same hand, losing with Ace's to a player with nines. Coincidences do happen!

 

[JoeTheJuggler]

I still think you needed to state your N up front. Why analyze the data now? Why not a month ago or a month from now? The potential for post hoc hypotheses is open when you do it this way.

I would expect that if he is making decisions such as described here, the results will revert to the norm, over time. And just as its rather rare to spot a run of 10 hands where he's only won one, play enough of these sequences, and he'll have one where he wins 9.

Hopefully soon!

So you continue the study until you get the result you're after? You really need to figure out the size N you need to have the power to answer the question you're after and state it up front.

Also the switch from on-line poker to in-person poker means you are mixing data from two very different things.

Also there is the problem of bias in the data collection since such collection is not done blind (or objectively) but by someone with a pretty strong position. (I've pointed to evidence of the goat and sheep bias in data collection already, I think.)

 

[JoeTheJuggler]

Recomputing the probability with your suggested changes to the odds:

10 * 1/3 * 1/5 * 1/5 * 2/3 * 1/5 * 2/3 * 1/46 * 7/10 * 1/2 * 1/5 = 0.000018 = 0.0018%

If you don't feel this is indicative of consistently poor luck, let me ask you how you feel that 'poor luck' might be established? What data should be collected and how should it be evaluated? These are serious questions, as that is exactly what we are trying to establish is or isn't happening.

I thought all these questions were answered some time ago. First, you need to define what you mean by "luck". It seems to me it's just being used here as an alternative explanation when you find any result that doesn't line up with expected outcomes. But remember, streakiness in data is expected even in random outcomes. (We don't really expect a series of coin tosses to alternate HTHTHT!)

So at best, all you're asking is what size N is required to have the power to reject the null hypothesis (that results are due to chance). If you reject that null, you still haven't supported this ill-defined "luck" hypothesis anymore than you have supported the hypothesis that the IPU is affecting the outcomes.

ETA: Your 0.0018%, is just the Texas Sharpshooter fallacy. Deal out any 5 cards, and the probability of getting that exact hand is lower than the probability of being dealt a Royal Flush. You really have to define your hypothesis at the beginning.

 

[Antiquehunter]

On the AK vs 73 preflop, you're somewhere between a 60/40 and 65/35 favorite (I know it seems like it should be better than that) - depends on how the suits match up. You need a calculator to get it precisely. I used a calculator - assuming AK unsuited and 73 suited (and the AK doesn't take away from possible flush draws), you're 61.33 to 38.23 with a small tie percentage.

Regarding the 1-outer situation: I play a LOT of poker, live & online, and genuine 1-outer situations are very rare. I can think of about 20 such hands I've experienced - I've been sucked out on twice, and I've done the sucking out once. So - I've lost 1:46 shots twice, and I've beaten 1:46 once. In 20 tries (that I know of). Again - I personally don't think that is enough tests to determine if I am 'lucky' or 'unlucky' in those situations. Sure - I've felt darned unlucky when losing (usually huge pots in these situations) and whooped for joy on the one I won - easy to slip into confirmation bias on these more rare situations. A 1:46 shot isn't that much different than picking one number on a roulette wheel. If you walked into Las Vegas, and made ONE bet on ONE number on ONE spin, and happened to win, beating 1:38 odds, you would be 'lucky' but not unusually so - happens all the time.

For me - and I'm not a mathematician - I would say you have insufficient data. I know people have berated me for saying I think you need too MUCH data. But I really do not think that statistically, these 10 hands viewed in isolation is much of an indicator. Yes, I would agree that for these times, it looks like he has been unluckier than should be expected. If we ran a monte carlo simulation, and tried this sequence of 10 hands, I don't know how many standard deviations off the norm he would find the result - I suspect that getting REALLY 'unlucky' on that one hand skews the net result significantly. (Remove that one, and calculate the 'unlucky' quotient based on the 9 hands and you'll get a much 'flatter' result). Anyways - I think you'd find that he's a couple of deviations off the norm, but not an 'outlier'. And this also doesn't show that he's consistently 'unlucky'. You would need to capture a number of these types of sequences to start building that sort of case to my mind.

And fundamentally - lets just say that over the next year or so, you show that in 8 of 10 sets of data, he's 2+ standard deviations off into the 'unlucky' realm - what hypothesis would you build around that? To suggest your hubby is a 'cooler' is a paranormal explanation - he could just have been 'unlucky' enough to have a rather unlikely run of cards.

I do hope it turns around for him soon - hopefully at the WSOP!

 

[Beth]

I still think you needed to state your N up front. Why analyze the data now? Why not a month ago or a month from now? The potential for post hoc hypotheses is open when you do it this way.

Actually, I did analyze it a month ago, I just haven't had time to post results here. I will also be analyzing it a month from now. The analysis method will not change though unless some error in how it is being done is pointed out to me. That's why I'm posting it here, so others can point out errors in the analysis method.

I don't agree with you regarding the need to state N up front. I am running a continuous analysis, recomputing the probability with each new addition to the database. I wouldn't object to going with a moving average (I do that with the races for a 25 hand average), but we don't have that much data for all the all-ins yet.

So you continue the study until you get the result you're after?

No. We continue this study of the data until we have either a better way to collect and analyze the outcomes he's reporting or until he gets bored with the project and stops bothering to collect data. Please recall that this was designed to assess how well his subjective observation was matching with reality - i.e. was his luck really as bad as he was claiming.

You really need to figure out the size N you need to have the power to answer the question you're after and state it up front.

Power isn't an issue because with a p-value of 0.000018 we can reject the null of random chance. Power refers to the probability of correctly accepting the null hypothesis. Power would be an issue if we were accepting the null and wanting to know how certain we were that the null was correct, and that would require a large sample size.

The p-value I've computed takes the amount of data available into consideration, and since it rejects the null, power isn't an issue and we can say with 99.99% confidence that the results are not consistent with random chance.

Also the switch from on-line poker to in-person poker means you are mixing data from two very different things.

Maybe. I'm not sure why those differences would have an impact on the outcome after an all-in occurs. Can you give a valid reason why they shouldn't be combined?

Also there is the problem of bias in the data collection since such collection is not done blind (or objectively) but by someone with a pretty strong position. (I've pointed to evidence of the goat and sheep bias in data collection already, I think.)

Yes you have. Blinding can not be done in this situation. However, the data being collected (the cards and outcomes) are objective. That's one of the appeals of it. I can see bias entering into it by not remembering to report the wins, but he swears that he is reporting all such hands. I believe him. Can you think of any other ways that bias might be affecting the results?

I thought all these questions were answered some time ago. First, you need to define what you mean by "luck". It seems to me it's just being used here as an alternative explanation when you find any result that doesn't line up with expected outcomes.

Yes, that's a fairly succinct way of putting it. Luck is when outcomes are significantly different than expected - i.e. the odds of what actually happens are statistically significantly different from what random chance would predict.

But remember, streakiness in data is expected even in random outcomes. (We don't really expect a series of coin tosses to alternate HTHTHT!)

Yes, I'm aware of that. That's why I multiplied the computation of the probabilities by 10 to account for all possible sequences of one win in ten contests. What I am doing is analogous to computing the probability of getting 3 heads and 3 tails rather than the probability of getting the sequence HTHTHT.

So at best, all you're asking is what size N is required to have the power to reject the null hypothesis (that results are due to chance).

The power of a test is the probability of correctly accepting the null hypothesis. The probability of rejecting the null incorrectly given the sample size is the p-value.

If you reject that null, you still haven't supported this ill-defined "luck" hypothesis anymore than you have supported the hypothesis that the IPU is affecting the outcomes.

In this case, rejecting the null is confirming that his observations were correct, that he is consistently losing such contests more often than random chance would expect. Whether you want to call it 'luck' or the blessings of the IPU, I think it does support the hypothesis that his outcomes are not in line with random chance.

ETA: Your 0.0018%, is just the Texas Sharpshooter fallacy. Deal out any 5 cards, and the probability of getting that exact hand is lower than the probability of being dealt a Royal Flush. You really have to define your hypothesis at the beginning.

It is, in fact, one fourth of the odds of getting a royal flush and exactly the odds of getting a Royal Flush in a particular suite. That is not the computation I am making. See above.

The reason I'm posting the results here are to work out any problems in my analysis. There is a constraint that only data from actual games will be used, which means that blinding can't be done. That's why we are only collecting data on the results of all-ins which can be compared to the predicted probabilities for those hands. At that point other factors should not affect the outcomes. If there is an error in the analysis, I appreciate having it pointed out and told how to correct it. For example, I'm pleased to be able to revise the computation using Antiquehunter's assessment of the probabilities of the various outcomes.

If you don't agree with my computation of the odds, how do you think it should be computed? Telling me I'm doing it wrong doesn't help if you don't provide a way to correctly compute the probability.

 

[Beth]

On the AK vs 73 preflop, you're somewhere between a 60/40 and 65/35 favorite (I know it seems like it should be better than that) - depends on how the suits match up. You need a calculator to get it precisely. I used a calculator - assuming AK unsuited and 73 suited (and the AK doesn't take away from possible flush draws), you're 61.33 to 38.23 with a small tie percentage.

Thanks for the correction. The revised p-value is 0.000022 or 0.0022%

For me - and I'm not a mathematician - I would say you have insufficient data.

We'll continue to collect more. As a statistician, I'm always wanting more data. On the other hand, I constantly work with minimal data sets because the analyses I do professionally are on very expensive tests. Every additional data point costs addtional $100's. The major problem with small datasets is low power, which means that if you don't reject the null, you don't have much confidence that the null is correct. But rejecting the null is done with a confidence of 1 minus the p-value regardless of sample size because the sample size is included in the computation of the p-value.

If we ran a monte carlo simulation, and tried this sequence of 10 hands, I don't know how many standard deviations off the norm he would find the result - I suspect that getting REALLY 'unlucky' on that one hand skews the net result significantly. (Remove that one, and calculate the 'unlucky' quotient based on the 9 hands and you'll get a much 'flatter' result).

Removing that hand and the p-value for the remaining nine goes all the way up to 0.000894 or 0.0894%. That change takes the confidence level to reject the null all the way down to 99.93%

For a normal distribution, the p-value of 0.000894 is equivalent to -3.12 standard deviations below the mean and the p-value of 0.000022 is equivalent to -4.09 standard deviations below the mean. While this isn't a normal distribution (it's better described as a series of bernoulli trials with varying probabilities) these computation should serve to give you an idea of how far from the norm those outcomes lie.

Anyways - I think you'd find that he's a couple of deviations off the norm, but not an 'outlier'.

Being more than three standard deviations from the mean is a fairly standard criteria to define outliers.

And this also doesn't show that he's consistently 'unlucky'. You would need to capture a number of these types of sequences to start building that sort of case to my mind.

That understandable. You're only seeing the recorded results. But from my hubby's POV, it's confirming what he suspected previously but did not have data to refute my contention that it was just confirmation bias.

And fundamentally - lets just say that over the next year or so, you show that in 8 of 10 sets of data, he's 2+ standard deviations off into the 'unlucky' realm - what hypothesis would you build around that? To suggest your hubby is a 'cooler' is a paranormal explanation - he could just have been 'unlucky' enough to have a rather unlikely run of cards.

I'm not sure what hypothesis to build around it. I was expecting a far more normal outcome, not what we're seeing. If what I expected had occurred, I had hoped to use the data to talk him into adopting a more positive attitude about such things.

I do hope it turns around for him soon - hopefully at the WSOP!

Thanks. I appreciate the sentiments although he had no plans to enter any tournaments of that sort.

 

[Senex]

As every question in life it comes down to defining your terms. What is the definition of luck.

In my own personal life I met a girl when I was young who didn't mock my naivete and taught me what I needed to know. Without her I might not be the gift to women I am today.

Luck, or cosmic intervention.

 

[jt512]

Let me bring you up-to-date and see if any one has an explanation for these results other than random chance.

He's had 10 all-in's and lost nine of them. Here are the hands:

He went all in after the flop with AK. His opponent had 3, 7. We estimate the probability of loss at 1/3. The flop came 3, 8, 7. He lost.

10, 10 against 5, 8. We estimate the probability of loss at 1/3. He lost.
K, K against Q, Q. We estimate the probability of loss at 1/5. He lost.
K, T against A, T. We estimate the probability of loss at 2/3*. He lost.

A, A against 9,9. We estimate the probability of loss at 1/5. He lost.
A, T against A, K. We estimate the probability of loss at 2/3. He lost.

The following hand is the only non-pre-flop all-in. In this case, he had
A, 6 against K, 3. They went all-in after the turn with 6,3,6,3 showing. We estimate the probability of loss at 0.02. He lost. The river was a 3.

J, J against Q,9. We estimate the probability of loss at 1/5. He won this hand!
A, K (suited) against Q, Q**. We estimate the probability of loss at 1/2. He lost.

A, A against 9,9***. We estimate the probability of loss at 1/5. He lost.

First of all, your probabilities of losing these match-ups are inaccurate. If I had a a hundred bucks for each time you've ignored Antiquehunter saying that these probabilities should be calculated by using a poker computer, I could bankroll my own casino. I ran two of your hands on this online holdem calculator, and if it is accurate, then both of your calculations were off, one drastically.

For TT vs 58, you estimate a loss probability for TT of 1/3. The calculator says 0.1643.

For AKs vs QQ, you give a loss probability for AKs of 0.5. The calculator gives 0.47 if neither Q is of the AK's suit, and 0.45 otherwise.

This brings me to my next point: your data aren't accurate. The suits of the cards matter, as the above paragraph shows. Your husband should be keeping track of the exact cards shown up.

My computation of the probability of getting one win out of those ten games as 0.00003.

This was computed as 10 * 1/3 * 1/5 * 1/3 * 2/3 * 1/5 * 1/3 * 1/50 * 4/5 * 1/2 * 1/5.

The above calculation is wrong. If we play n hands, each with a probability of losing of p_i, i=1...n, and X is the number of hands lost, then

which, using your data, I calculate to be 0.000056 (someone should check my arithmetic).

To calculate a one-sided p-value, we have to add to this the probability of losing all 10 hands, which nudges the probability up to 0.000057. This is about double the probability you calculated.

However, until this post, you made the point repeatedly that you were only using all-in pre-flop hands. Yet, in this calculation, you mysteriously have included an all-in post-flop hand. This was not part of the original hypothesis. You've changed the rules of the experiment part way through it to include selective bad beats that don't qualify for inclusion. If we exclude the post-flop bad beat, the revised p-value is 0.0009; that is just under 1 in 1000, and is starting to look more like an ordinary bad run than a near impossibility.

** This qualifies as a race. Our statistics are now 22 wins out of 57 races which has a p-value of .0556.

So, when you stick with the original hypothesis, your results still are not significant at the 0.05 level (one-tailed).

Mathematically, you are testing the hypothesis that the probability of your husband wining is less 0.5 for showdowns of the type you've described earlier. Let's say that you continue collecting data for another year, and stop the experiment according to some rule not related to the calculation of an interim p-value (arguably, that's impossible at this point, but let's ignore that). Furthermore, let's that say that the final calculated p-value is 10^–8, overwhelmingly rejecting the null hypothesis. What could this mean?

1. The null hypothesis is correct, and an extremely unlikely event occurred by chance.

2. You made a methodological error in constructing the experiment such that the probability of winning the type of hands that were to be counted was actually less than 0.5.

3. There were errors in data collection.

4. There were errors in the analysis.

5. Your husband was cheated.

6. There is a supernatural force—call it luck—that negatively affects your husband while playing poker.

It should be evident that no matter how firmly the null hypothesis is rejected, the possibility that one of the first five possible explanations above (and probably more I haven't thought of) is responsible for the result is astronomically more likely than the sixth.

 

[Beth]

This brings me to my next point: your data aren't accurate.

The data are accurate; the probabilities are estimates. I'm more than happy to change them when someone provides me with a better value. I haven't used a poker calculator because I haven't been able to figure out how to use them to help me estimate the probabilities I need. As you point out, without the suites, the probabilities are not completely accurate.

The suits of the cards matter, as the above paragraph shows. Your husband should be keeping track of the exact cards shown up.

I agree, but I'm not the one collecting the data. If he chooses not to keep track of suites, that is his choice not mine.

The above calculation is wrong. If we play n hands, each with a probability of losing of p_i, i=1...n, and X is the number of hands lost, then

[qimg]http://jt512.dyndns.org/images/p1.png[/qimg]​

which, using your data, I calculate to be 0.000056 (someone should check my arithmetic).
To calculate a one-sided p-value, we have to add to this the probability of losing all 10 hands, which nudges the probability up to 0.000057. This is about double the probability you calculated.

Yes, that is a more accurate formula. I've been somewhat lazy about that computation, multiplying by 10 rather than computing out the probability that each hand was the only winner. Thanks for making the computation. I'll try to check your arithmetic laterWe are now down to only a 99.4% confidence level for rejection of the null hypothesis.

However, until this post, you made the point repeatedly that you were only using all-in pre-flop hands. Yet, in this calculation, you mysteriously have included an all-in post-flop hand. This was not part of the original hypothesis. You've changed the rules of the experiment part way through it to include selective bad beats that don't qualify for inclusion.

Actually, what we did was start a new experiment based on feedback I had received here - to wit, that when the all-in call is made, the probabilities can be computed and it doesn't matter whether it's pre- or post-flop. Hands that fit the original criteria are added to our previous dataset, but we have also started building a new dataset with the criteria for inclusion being just that it was an all-in hand.

If we exclude the post-flop bad beat, the revised p-value is 0.0009; that is just under 1 in 1000, and is starting to look more like an ordinary bad run than a near impossibility.

Excluding data just because you don't like it isn't kosher. Incidently, a p-value of 0.0009 is not what I would term an ordinary bad run. It would allow us to reject the null hypothesis at a confidence level of 99.91%.

So, when you stick with the original hypothesis, your results still are not significant at the 0.05 level (one-tailed).

Correct, the p-value is slightly above 0.05% for the 'races'.

Mathematically, you are testing the hypothesis that the probability of your husband wining is less 0.5 for showdowns of the type you've described earlier. Let's say that you continue collecting data for another year, and stop the experiment according to some rule not related to the calculation of an interim p-value (arguably, that's impossible at this point, but let's ignore that). Furthermore, let's that say that the final calculated p-value is 10^–8, overwhelmingly rejecting the null hypothesis. What could this mean?

It means that my dh's complaint seems based on an accurate assessment of his results, not confirmation bias due to his remembering the bad beats and not the wins.

1. The null hypothesis is correct, and an extremely unlikely event occurred by chance.

Yes, that's one possibility, as it is for any experiment. The p-value tells us exactly what the that probability is for the actual outcome. However, it's generally not considered a reasonable conclusion that a p-value of 0.001 or less was due to random chance.

2. You made a methodological error in constructing the experiment such that the probability of winning the type of hands that were to be counted was actually less than 0.5.

Yes. That is why I started this thread and have posted our results for discussion. The feedback I received for the 'races' experiment was that the probability of 0.5 was a reasonable estimate of that probability.

3. There were errors in data collection.

This is always a possibility. It's certainly a reasonable supposition on your part. However, this is not an explanation for our results that I or my spouse finds convincing.

4. There were errors in the analysis.

Quite possible. Again, that's why I'm posting the results here. Thanks for your help in this regard. I'll make the corrections and let you know if I get agreement with your values. However, the corrections don't appear to make much difference in the interpretation of the p-value. It's still very low.

5. Your husband was cheated.

I actually find this to be quite unlikely given the circumstances of the games he plays. With his buddies, the composition of the other players changes with every game and there are only a few people who are there every time and there's no one person who wins consistently over time. He's been playing with them for years. The 'races' experimental data consistent primarily of on-line play money games where cheating seems a very unlikely possibility. Again, while it is a reasonable supposition for you or others reading this, it is not an explanation for our results that I or my spouse finds convincing.

6. There is a supernatural force—call it luck—that negatively affects your husband while playing poker.

I see no reason to posit a supernatural force. I don't define 'luck' that way.

It should be evident that no matter how firmly the null hypothesis is rejected, the possibility that one of the first five possible explanations above (and probably more I haven't thought of) is responsible for the result is astronomically more likely than the sixth.

I agree. Still, the data are consistent with a lower than normal win rate for all-in show-downs, which is what he has been complaining about. I've been working to correct any issues with regard to reasons 2 and 4. Reasons 3 and 5 are not reasonable conclusions for us. Reason 1 is not a reasonable conclusion with the p-values we have. So I'm looking for other possible explanations and we are continuing to collect data to see if his win rate will eventually approach something more in line with the expectations of random chance.

 

[Antiquehunter]

I'm not sure if its a valid mathematical principle, but one issue that is skewing the results in this particular sample of 10 hands, is the issue of the 1-out hand (that happened to be captured post-flop).

As I mentioned before, this is an extremely rare occurence in and of itself. The circumstances that lead to such a hand where only one out can save a player are highly irregular and do not come up very often. Whereas the other all-in conflicts are all quite common.

I'm trying to find an appopriate analogy, but can't readily craft one. My point is that you have a bunch of fairly common-place and not especially rare occurences (the biggest 'edge' being about 20:80) and then all of a sudden, you include what is in and of iteslf, a comparatively rare situation, about 1:46. This event itself can really skew the result in such a short run of events. Much like with judges at subjective events, they dump the top & bottom numbers.

I don't think that it matters that the measurement happened pre or post flop, assuming the data is gathered correctly. As I mentioned earlier in the thread, once all the money is in the middle, no more skill is involved in the hand, its purely mathematical, if the all-in is called, who has the mathematical edge, and whether or not one beats the odds or not as the cards are turned.

But my argument is that the one hand specifically should be thrown out because it has such a huge potential to only impact the calculation in one direction. IF the hand had played out as mathematically dictated (ie - your husband didn't get 'unlucky' and his 45:46 edge brought him the pot) then there would be little impact to the calculation. IF he gets 'unlucky' then it has a big impact on the rest of the calculation. Someone more educated in math probably has a principle that suggests why it should be treated separately.

 

[jt512]

I'll try to check your arithmetic later.

I made a data entry error! I was using 1/5 instead of 4/5 for the one win. Using the correct term, I get a p-value of 0.00022 for all 10 hands. When the post-flop hand is excluded, the p-value is 0.003, which is definitely within the realm of a bad run.

 

[jt512]

The feedback I received for the 'races' experiment was that the probability of 0.5 was a reasonable estimate of that probability.

Then you either received bad feedback, or you misinterpreted the feedback you got.

If you are trying to figure out whether your husband's results differ from the expected result by chance (or "luck"), you have to remove the element of skill from the calculation. However, assuming that he has a 0.5 probability of winning each pre-flop showdown (or equivalently half of such showdowns in the long run) implies that the quality of his pre-flop decisions are exactly average (in terms of probability of winning as opposed to expected value). Why else would he be expected to win exactly half the time? Clearly not all pre-flop hands have the same probability of winning, and therefore a player's decisions about which hands to go all-in with and call with will determine the long-run proportion of these showdowns that he will win.

Although I think this is obvious, for illustration, consider two players. The first player only goes all-in with AA. Since this is the best pre-flop hand in the game, his opponent will never have a better hand than him, and he will win the showdown well more than 50% of the time (87% according to the online poker calculator I posed the link to up-thread). In contrast, consider a player who will go all-in with QJs or better. He will usually be behind when called (I think), and will thus win less than 50% of these showdowns. Thus a player's preflop behavior—his skill—matters, and you can't just assume that a particular player will win 50% of the time.

If you want to eliminate skill from the hypothesis, you can do so, but you have to calculate the probability of winning for each hand separately. That is, you have to calculate the conditional probability of winning the hand (conditioned on what the player's and his opponent's hands actually are). Then, it doesn't matter what cards the player habitually goes all-in with or calls with pre-flop. The player who only goes all-in with AA will still have a higher expected win rate than the player who goes in with QJs or better, but you will be comparing that player's results with his specific expected win rate.

 

[Antiquehunter]

@JT - The original hypothesis from Beth was that her husband was 'unluckier' than could be expected only in 'all-in' situations. As such, I pointed out that once all the money was all-in (whether it was skill or foolishness that arrived is moot) - one can determine simply the mathematical expectation of that situation.

ie - AhAd vs QsJs preflop - that has a specific EV. (80.12% for AA, 19.52% for QJs) Now whether or not skill was involved in getting to this stage is immaterial. Over the long haul, the player with the AA should win about 80% of the time.

At first the test data was only going to include 'race' type of hands. These hands vary only slightly in value (generally 52/48 range). But then we looked at a broader variety of hands. Beth & the player didn't feel that running out hands around the kitchen table was proof enough - hands that came from a 'real' game (either online or live) were seen as necessary to test this 'luck' factor.

Anyways - that is how Beth got to this point. Just to condense the thread for you.

 

[jt512]

@JT - The original hypothesis from Beth was that her husband was 'unluckier' than could be expected only in 'all-in' situations. As such, I pointed out that once all the money was all-in (whether it was skill or foolishness that arrived is moot) - one can determine simply the mathematical expectation of that situation.

Yes, but you actually have to calculate that expectation. You can't assume that it's 0.5, for reasons I explained above.

ie - AhAd vs QsJs preflop - that has a specific EV. (80.12% for AA, 19.52% for QJs) Now whether or not skill was involved in getting to this stage is immaterial. Over the long haul, the player with the AA should win about 80% of the time.

Right, but 80% is a long way from 50%. Thus a player who only ever goes all-in preflop with AA will expect to win 80% of the time when he goes all-in, not 50%. Therefore, the long-run percentage of these showdowns that a player wins depends on the distribution of hands he goes all-in with. It cannot be assumed to 50%.

At first the test data was only going to include 'race' type of hands. These hands vary only slightly in value (generally 52/48 range).

That would be the only possible "justification" I could see for assuming a constant expectation of 50% for every hand. We can do a sensitivity analysis to see the possible range of error on the calculated p-value caused by this assumption. Beth's data currently consist of 22 wins in 57 "races." The correct one-tailed p-values for expected wins of 52% and 48% are 0.029 and 0.098, respectively. Whether this is a little error or a lot is a matter of judgment; I'll only point out that at the ubiquitous 0.05 level of significance, the null hypothesis is rejected in the first case, but not in the second.

Jay

 

[Gr8wight]

I'm not the one collecting the data. If he chooses not to keep track of suites, that is his choice not mine.

So, all the data you have presented us to date is merely anecdotal. Without a careful log of each and every hand played eliminating confirmation bias is impossible. Have you considered the possibility that your husband is just not that good a poker player?