The Law of Large Numbers. How Large?

Question

My questions is very simple. How many hands are required before the usual statistics are realized in reality? For example, I know that a flush draw pays off roughly 1 in 3 times. So if I get pot odds of better than 2 to 1 I am going to make money in the long run. My question is what is the long run. I have chased about 10 flush draws with proper pot odds over the past few months ( I am only a weekend player) and lost on all of them. How many flush draws do I need to play to make money assuming I am getting 2 to 1 pot odds? 1000? 1,000,000? 500? Is playing with knowledge of pot odds actually quite useless if I am not playing enough hands for these odds to be realized?

Answer 1

I don't think I would look at is like that.

Let's say you are 1/3 to hit the flush.

The chance of not hitting is 2/3.

Not hitting twice in a row 44.44%

44.44%  2
29.63%  3
19.75%  4
13.17%  5
8.78%   6
5.85%   7
3.90%   8
2.60%   9
1.73%   10
1.16%   11
0.77%   12
0.51%   13
0.34%   14
0.23%   15
0.15%   16
0.10%   17
0.07%   18
0.05%   19
0.03%   20

There is 1.73% chance you don't hit 10 times in a row. That is just plain unlucky but you should still play the numbers. There is a lot of variance in poker and that is why you need a bankroll to absorb it.

To put that in perspective the chance of pocket aces is 0.45%.

In 100,000 draws a losing streak of 25 and win streak of 10 would not be uncommon.

You just plain need to play more hands. Go with lower stakes games where you can afford some losses as you learn the game.

If you are playing a home game you can run the board multiple time to reduce variance.

By saying 1/3 indicates a hole in your game to me. You are only 1/3 if you are getting two more cards. You are only getting two more cards for that price if you are all in. I seriously doubt you had 10 all ins on the flop and 9 re-buys. If you are calling a flush draw on the flop because you are getting 2:1 then you will continue to lose money. You can lose to a higher flush, boat, and quads and unless it is an all in you will face another bet.

On the flop you have 9 outs and 45 cards to come. Your immediate hand odds are 37 / 9 = 4.22 : 1. Unless you are all in or all you opponents are all in then you need pot odds of 4.22 : 1 and an unpaired board. Yes you might get more money in the pot if you hit for better implied odds. But if you don't hit you are going to be facing another bet on the river. Pot odds are not useless. I suspect you are not playing proper odds and you are exploited by players playing proper odds.

Answer 2

You're actually making several questions:

My question is what is the long run

Short answer: is long as hell.

Long answer: I'll talk about shorthanded hold'em plain games. For tournaments, I think the variance is even higher.

I read and listen a lot that 500k hands played in the same level and game is a fair bet to tell if you're a winner or not, and after studying a lot of databases of winner and losers players I think it's a reasonable amount. I've seen lots of 100k hands bad runs of winner players, but for a regular winner players that is a consistent winner in millions of hands, to have a 500k hands breakeven or loser run is really difficult and I haven't see any, and I saw a lot of graphs of winner players. Instead, breakeven or loser runs of 100k hands on winner players are relatively common (It also depends a lot on winrate. A player with a .5 BB/100 winrate will face several really long losing runs per every million, while a 7 BB/100 winner likely not. )

This doesn't mean that after 500k hands played you can be sure your real winrate is near your 'theorical' winrate. I've seen BB's of difference on the same player in runs of a million hands for the same type of game.

Also, to play several hundreds of thousands of hands you need a lot of real time, and your skill itself and the metagame do change in that time, so the idea that you can play so many hands that you can annulate the variance is just a desire, but not a real possibility.

How many flush draws do I need to play to make money assuming I am getting 2 to 1 pot odds? 1000? 1,000,000? 500?

You can never tell how many. You can just calculate the odds of not winning money after x events. The answer from Paparazzi explains this well.

Is playing with knowledge of pot odds actually quite useless if I am not playing enough hands for these odds to be realized?

Well, it depends what you understand for useless. If you're playing not many hands a month, you can't actually win or lose too much stacks; also, the difference between playing better or worse, in terms of money, is diminished by the fact you're playing not many hands. However, do a bad move on purpose is irrational unless you find lots of fun doing it.

for instance, if you have to face a push having a straight project, and lose 1$ on call by average, doing that just 100 times a year just cost you 100$ on average. It's up to you to decide if that is relevant or not, but sure playing against the odds won't make you money.

Disclaimer: I've used to be a power player several years ago, playing NL200$ and 400$. I'm talking 'bout those days. If the metagame is harder now than then -which I believe is the case-, edges are smaller and bad runs are even longer. Also, if you're playing microlimit games, its exactly the opposite, and be a loser after 80k/100k hands is a strong signal that you're doing a ton of errors.

Answer 3

My question is very simple

I wish I could give a simple answer of X hands, and indeed if you want an easy answer, I'd say "a lot" or "many thousands of hands", but I think it's really beneficial to understand why it can be a little more complicated than "X hands". (Note: the following is super informal, so terms like variance aren't always meant to be taken in a precise,technical way)

1. What would you consider to be an acceptable deviation from the expected mean to call something "normal". Take the ever handy example of flipping a coin, perhaps 100 trials where you win if the outcome is heads. Would 40 heads be close enough to the expected mean (obviously 50) to say you ran normal? 45? 48? Only exactly 50? Because of the law of large numbers, you should get closer and closer in general to that threshold of 50% as you keep flipping but you could go literally 10's of thousands of flips without hitting the mark or going above, so is 49.5% OK?

2. In poker you aren't flipping coins. In your example, you're trying to hit a flush draw which we might assign a rough probability of 1/3. There are also draws with higher or lower probabilities. The more unlikely an event is of occurring then the more variance you'll see in outcomes over the trials you have, so something like the outcome of X flush draws has more "volatility" than something like the outcome of X coin flips. Increased variance leads to a longer time that it will take for the law of large numbers to have the same impact.

3. In poker, winning or losing a hand doesn't ultimately mean much in the long term when compared to how much you won or lost. Hitting one flush draw to win $100 wipes out 10 hand flush draws that didn't hit where you lost $10. You can think of pot size as another factor that increases the variance of your outcomes because what you really should be interested in over the long run is how your $'s won/lost compares to the expected $'s won/lost.

4. To fuse the last couple points together, you won't win anything (or lose anything) over the very long term if you always have the exact right odds to make a call, for example. You'll instead end up somewhere close to your expected long term value of breaking even. What's important is how big of an edge you have in your hands. If instead of 2 to 1 pot odds for your flush draw, you're receiving 3 to 1 odds, your expected value will be higher and you actually need to win fewer times to break even. What this does is effectively increase the trajectory of your win/loss rate compared to the breakeven rate that's you'd see after enough time because of the law of large numbers. Increase your edges (for example, better pot odds) and that trajectory increases. Make bad calls or plays with negative expectation and that trajectory declines. The sharper this trajectory is, the sooner your long-term outcomes will "converge" to something that's above the long-term breakeven expectation.

5. There many, many scenarios that come up in poker--the odds of winning and pot odds you have and size of pot is rarely duplicated. What should matter is the aggregation of all the hands we play, the "edge" that's aggregated through those hands, and the aggregate outcome ($'s won/lost). When everything is taken together, it can be difficult to see whether you ran hot or ran cold compared to expectation over any period of time.

6. Contrary to the last point, you could simplify things to say exactly flush draws and whether or not you ran good over a sample of 10. But it's human nature to compartmentalize these things and remember the negative experience. You might not realize the period of time where you hit 6 out of 10 flush draws. Or during this bad streak of missing flush draws, you may not have noticed that your ran good hitting straight draws or flopping sets or being dealt high pocket pairs. There's so many possible scenarios that we could keep track of, but since there's so many there is bound to be anomalies over time that stick out to us, especially when bad things happen in big pot. Overall though, things should tend to balance each other out over a long period of time.

7. There's a ton of variance involved in the game. The only things we can do to reduce the effect of that is to either play smaller pots or only proceed with hands where your edge is larger. People with smaller bankrolls may take this approach and there may be times to use in in tournaments where your risk of being eliminated isn't worth pursuing a tiny edge. Some people with large bankrolls embrace the style of pushing every edge, no matter how small, and then understand that there will be large up and down swings to their bankroll.

8. So are pot odds useless if it takes a tremendous amount of time to ever converge to your "normal" expected winrate? No. Unless you have psychic abilities of knowing what cards are dealt next, it's ALL that we have to make the best choice possible. Playing willy nilly without using odds will send you on that downward long-term trajectory below the breakeven expectation and you will almost surely end up a loser. Bad players can get lucky for a few hands. They can get lucky for a lot of hands and sometimes it seems like they time their luck for only the biggest pots. But most quietly lose and go unnoticed and even the ones that start out lucky lose it all back over a long enough period of time.

Answer 4

Check out this great statistical analysis of poker. Essentially, the true skill of a player becomes apparent somewhere in the order of 10,000's hands:

Poker is a game that happens in the long run. The results on a per hand basis are too affected by variance to rely on them as accurate predictors for the level of skill a particular player has. These results are also too short term to identify trends a particular opponent has. It is only after tens of thousands of hands when you see the short term variance become less significant than a player’s skill.