Do you believe in Luck?

[Beth]

Okay, here's the situation. My husband's a poker player. He likes the Texas Hold-em game and plays quite a bit on-line and a couple of times a month in a garage or basement game. He's constantly going on about his bad luck, how he never seems to catch the right card no matter what the odds.

I say, you're focusing on the bad beats and not paying attention to the wins. He swears his luck is phenomenally bad. I'm a professional statistician. I tell him to collect data and I compute the probability. So he did. And I did. And he has had consistently bad luck! Given the number of hands he's collected data on, the probability of luck as bad or worse than that is less than 10%. He is running fairly consistently at 40% wins, both overall and for a running 25 game average, for a situation that should be about 50/50.

In the next paragraph, I'll explain the data collection and analysis we're doing. I'd appreciate any suggestions for improving either.

He is only collecting data on one particular type of hand. A showdown situations where he and one other player are All In before the flop. In addition, he's only looking at a the outcome when one of the two sets of two cards is a pair and other person had two over cards. The probability of winning is approximately 50/50. (It's actually more like 48/52 but so far, I've just been computing the odds at 50/50). He terms this a 'race'.

He started collecting data back in March. He was just keeping a running total of how many 'races' he was in and how many he won. He is currently at 21 wins out of 54 races. Assuming a 50/50 probability of winning races, the probability of wining 21 or fewer out of 54 is 0.0668.

He been meticulous about recording outcomes of all such races win or lose. I've recently talked him into recording what the actual cards were that he won and lost with, so I should be able to start computing the probabilities more accurately.

He is, at least, feeling vindicated regarding his complaint about bad luck.

 

[Antiquehunter]

I'm not sure your sample size is large enough to be meaningful. Assuming we treat all races as a 50/50 prop (as you point out, they're 52/48 or in that range, depending on the exact cards, straight / flush possibilities etc...) then he 'should have' won 27/54. He is 6 off that mark - unlucky, but not extremely so. Collect a couple of hundred samples and see if he heads more towards the mean.

One other consideration also, if you're gathering this information when you're multi-way, the odds are a LOT different. (A pair vs TWO hands goes down in value, unless the two hands take away some of each others outs.) Finally - if you're playing at a free-money table, where there is a lot of pre-flop action, you may further be able to make some assumptions. For example, in free-money play, you see people going all the way with an Ax. As such if there is a lot of preflop action, I would expect that it is likely there are more aces than normal in people's hands. Presumably then, with 'dead' aces, I would tend to push my larger pairs stronger. Even if only called in one place by an AK, I would expect it likely if there were lots of 'limpers' that someone else folded an Ace. Can't be proven of course, but just a thought.

As a serious poker player myself (made the money at the WSOP) my advice is that if he considers himself to be a more skilled player at the table than average, then he should try to avoid race-off situations except where his chipstack in a tournament, or in a cash game, suggest this is the best value for money.

I know you're testing his 'luck' in a specific gaming environment, but there are poker hand simulators you can download where he could run 10,000 hands of QQ vs AK very quickly, to see if his results are in line with the true mathematical odds.

-AH.

 

[Loss Leader]

The data appears to be flawed. You're collecting data long, long after a lot of human intervention has taken place. Every bet and every raise, every person who folded, affected the outcome. Your husband may just be preternaturally bad at knowing when to fold. In fact, it would appear that all you've proven is that he's way, way too agressive overall.

Go back to start and record his hole cards for twenty hands - no matter how those hands turn out. Are his hole cards statistically worse than chance? I would bet that they are not.

Now, when he has a pair of nines with a Nine, King and Ace showing after the flop, and the guy to his left bets 100 and the next guy goes all in with 800, does he call, raise, or fold? If he says the answer is anything other than fold, he probably should stop playing poker and find a different hobby.

 

[JoeTheJuggler]

I think it depends on what you mean by "luck". In the "races" described, you wouldn't expect outcomes to line up with the odds perfectly unless you had a really large dataset. Even so, within that dataset, we would expect there to be streaks where one side wins more often than he "ought" based on the odds.

By the way, I would describe these nearly 50-50 situations as "coin tosses" rather than "races". There is no skill at that point. And even tossing a fair coin, we don't expect the result of 20 tosses to be HTHTHTHTHT. . .

 

[JoeTheJuggler]

The data appears to be flawed. You're collecting data long, long after a lot of human intervention has taken place. Every bet and every raise, every person who folded, affected the outcome.

I'm willing to accept the premise that at that point, whatever happened before, the outcome of the all-in heads up play is a coin toss (both hands have a nearly 1:2 chance of winning). In fact, in such games, the cards are turned up, and we can actually calculate the odds with the information in front of us. (We know how many cards remain in the deck and how many of them will win for one player or the other.) So I think at that point, we can indeed ignore what happened before.

 

[Akhenaten]

You've got to know when to hold 'em . . . etcetera.

Lucky players don't win at poker - skillful ones do. And vice-versa.

 

[Fredrik]

Most poker players use software for that sort of thing. Hold'em Manager is the most popular one. You can compare the graph of your actual result with the graph of your "all-in EV". The latter is calculated by replacing your actual results in the hands where you were all-in on the turn or earlier with the statistically expected results. This way you can clearly see if you've been lucky or unlucky in those situations. However, it's unlikely that those graphs give you the right idea about how you've been running. I just checked my database for May and June, and it looks like less than 0.6% of the hands I played were all-in on the turn or earlier. (I play cash games. Tournament players end up all-in much more often).

Because of this, I really think that the money won/lost is a much better measure of how lucky or unlucky he's been than these all-in EV graphs.

Back in 2008 I had a couple of months when I was about 50 buy-ins (i.e. 5000 big blinds) below expectation in about 80K hands. I've seen a graph (posted at a poker forum) that was 300 buy-ins below expectation. Somehow that guy still managed to break even, so he must have been running very well in the other hands.

It's definitely possible to run insanely bad over 100K hands or so. It feels like your bad luck is an entity that follows you around. It's extremely frustrating, and it's impossible to continue to play well under those circumstances.

I don't know what to make of the poll question. I think you would have to explain what you mean by "luck" for the question to make sense. I believe that if a million people buy a $10 ticket for a lottery that, by the rules of the lottery, awards 50% of that money to a single winner, selected using a random number generator, then someone will win $5M. Does that mean I should answer "yes"?

 

[JoeTheJuggler]

Lucky players don't win at poker - skillful ones do. And vice-versa.

True enough, and it's also true that someone skilled in martial arts or boxing can beat someone who is less skilled but bigger and stronger. [ETA: But if they're both equally skilled in martial arts, the bigger or stronger one has a distinct advantage.]


However, if both persons are equally skilled (or we ignore all situations where skill matters), the odds still don't tell us that you will win exactly half of all 1:2 chance gambles. There will indeed be streaks of various lengths. These streaks of course don't change the odds of subsequent instances--they're still 1:2.

No one is inherently lucky, but there is certainly an element of chance in poker. (You can see the top skilled players going against each other. It generally comes down to who gets the cards, even though they'll very consistently destroy less skilled players.)

 

[I Ratant]

"luck" is when preparation meets opportunity.
Skilled (prepared) players win at cards... or anything else.
Unfortunate happenstances can beat anyone at any time, though.

 

[JoeTheJuggler]

"luck" is when preparation meets opportunity.
Skilled (prepared) players win at cards... or anything else.
Unfortunate happenstances can beat anyone at any time, though.

But again, unless I'm grossly misreading the OP, she's asking us *only* to consider situations where all play is finished except the dealer turning cards up--there's absolutely nothing either player can do to change the odds or outcome, and at that point the odds for each player to win are even.

Again, if the premise is correct, we could as well be talking about a series of coin tosses. Even though the odds for heads and tails are equal, it's unreasonable to expect an exact sequence of THTHTHT. . . . And after 10 flips, not getting exactly 5 of each is no big surprise.

 

[JoeTheJuggler]

Most poker players use software for that sort of thing. Hold'em Manager is the most popular one. You can compare the graph of your actual result with the graph of your "all-in EV". The latter is calculated by replacing your actual results in the hands where you were all-in on the turn or earlier, with the statistically expected result. This way you can clearly see if you've been lucky or unlucky in those situations. However, it's unlikely that those graphs give you the right idea about how you've been running. I just checked my database for May and June, and it looks like less than 0.6% of the hands I played were all-in on the turn or earlier.

But again, if I'm reading the OP correctly, we already know the odds and then the actual results.

To me the question is, if the odds of these situations are 1:2, and you'd lost 7 out of 10 times (ETA: actual numbers: 21 wins out of 54*), is it meaningful or somehow significant?

My answer is, no. That result falls within the realm of what we'd expect due to chance.

ETA: If you've lost 90 out of 100 of these, or if you tossed a coin 100 times and came up with 90 heads, then I would suspect that it's not a fair coin or something else is up. (That is, I'd find that outcome significantly different from the hypothesis that the result is due only to chance.)

* And that's if the probability of winning were actually 50% and not something more like 48%. If the latter, his 21 out of 54 is almost a perfect reflection of the odds. Even so, we'd still expect streaks and wouldn't be surprised if in 54 trials we don't get exactly the mean number of wins for that probability.

 

[Antiquehunter]

I made the assumption you're treating these races as all-in pre-flop. If he's waiting post-flop and pushing 99 with overcards on the board, then as pointed out, you can throw the 'race' premise out the window.

For a pure 'test' of 'luck' run a simulation of 10,000 hands QQ vs AK pre-flop.

 

[Akhenaten]

<polite snip>

No one is inherently lucky, but there is certainly an element of chance in poker. (You can see the top skilled players going against each other. It generally comes down to who gets the cards, even though they'll very consistently destroy less skilled players.)

Acknowledged and thanks.

I ought to have been less lazy and said something along these lines myself.

 

[Fredrik]

But again, if I'm reading the OP correctly, we already know the odds and then the actual results.

Did it seem like I was talking about something other than that? My point was just that the hands for which luck is measurable is a very small percentage (0.6% for me apparently) of the total number of hands played.

(Edit: Beth was talking about preflop all-ins. In my database for May and June, those are less than 0.3% of the total number of hands).

 

[JoeTheJuggler]

I made the assumption you're treating these races as all-in pre-flop. If he's waiting post-flop and pushing 99 with overcards on the board, then as pointed out, you can throw the 'race' premise out the window.

For a pure 'test' of 'luck' run a simulation of 10,000 hands QQ vs AK pre-flop.

I think she did specify pre-flop. But the only difference pre- and post-flop makes is how complicated it is to calculate the probabilities (and thus ignore all hands in a similar situation when the odds of winning aren't 1:2).

From that point on (even pre-flop), it's heads up all-in, and there is no more skill involved. If we know the odds are 1:2, it is exactly the same as flipping a coin.

In other words, if I'm reading the OP correctly, we have eliminated skill. If his skill (or his opponent's skill) resulted in him going all-in (pre flop or not) with a better hand or a worse hand than his opponent, we're not counting that situation as a trial. So all that's left are the instances when his skill (or lack thereof) has resulted in a 1:2 chance of winning, and there's nothing left to do but watch the dealer throw cards.

 

[JoeTheJuggler]

Did it seem like I was talking about something other than that? My point was just that the hands for which luck is measurable is a very small percentage (0.6% for me apparently) of the total number of hands played.

But again, unless I'm mistaken, she's saying that of these 54 hands, 100% of them are this kind of hand. It doesn't matter, how well or poorly he has played in other hands. So I don't think she's interested in the calculation you're offering.

If I'm not mistaken, she's saying he's played 54 coin-tosses and has won 21 of them, and she's wondering if that's significant.

ETA: And just to clarify, when you say "the hands for which luck is measurable" do you mean "hands where at the point no more skill is involved the odds of winning are 1:2"? If not, then I'm definitely misunderstanding you. (This is partly why my first response was to ask for a definition of "luck". That's not really a statistical term.)

 

[Skeptic Ginger]

Depends on how you define 'luck'. If you define it as predictive, then no. If you define it as descriptive of what has occurred, then, yes it exists. The lot of each of us clearly is not equal. So assuming the latter, I said, yes.

 

[Antiquehunter]

Well post-flop, there is a lot more information available (3 out of 5 community cards dealt), and the odds of the AK beating QQ post-flop are dramatically different than 52:48 depending on the flop. (If the AK pair up, they're now a monster favorite. If they miss, then they are now a big dog.)

A pure test of 'luck' would be pre-flop all in, hand X vs hand Y. No more skill involved. Just happens that two overcards to a pair is pretty close to a coin toss, so it makes for an interesting test (I guess - not really interesting to me.)

 

[Fredrik]

But again, unless I'm mistaken, she's saying that of these 54 hands, 100% of them are this kind of hand. It doesn't matter, how well or poorly he has played in other hands. So I don't think she's interested in the calculation you're offering.

If I'm not mistaken, she's saying he's played 54 coin-tosses and has won 21 of them, and she's wondering if that's significant.

Right, but her husband is saying that he's been unlucky. I'm saying that she can't find out if that's true by examining a very small percentage of the hands he's played. I don't know why you're mentioning skill. I wasn't talking about skill at all.

 

[JoeTheJuggler]

Depends on how you define 'luck'. If you define it as predictive, then no. If you define it as descriptive of what has occurred, then, yes it exists. The lot of each of us clearly is not equal. So assuming the latter, I said, yes.

See my posts. I think she's asking, in a coin toss (no skill, 1:2 chance of winning), you've won 21 out of 54, is it significant? (I take "luck" or being "lucky" to mean that somehow the results are significant or surprising.)