Running bad affects me so I'm trying to understand the numbers underlying it rather than joining in on some fallacy or other. However, this is burning a hole in my brain so maybe some numerical genius here can set me straight?

Scenario. All-in Pre-flop. Winnings after rake 193.2bb if won, 0 if lost. No chopped pots. Probability of winning 60%.

The single instance is easy. The EV is 115.9bb but I get 193.2bb 60% of the time so I'm running good 60% of the time.

The two instance case seems easy. Options are win-win, win-lose, lose-win, lose-lose. The EV is 231.8bb but win-lose only nets 193.2bb so it's only win-win that means I'm running good which happens 36% of the time.

That's a strange result but I can see how the low number of instances could cause distortions so I whip out Excel and simulate up to 7 cases hoping that would be enough to see a pattern or at least something asymptotically trending somewhere. Results

1 instance p-value run good: 60%
2 instance p-value run good: 36%
3 instance p-value run good: 65%
4 instance p-value run good: 48%
5 instance p-value run good: 34%
6 instance p-value run good: 54%
7 instance p-value run good: 71%

Can someone please explain what's happening here? Is there a simple answer to where the trend will end up?

Thanks

there are variance calculators online where you input your expected winrate in bb/100 hands and standard deviation and it creates charts for you. basically, your winnings are a random walk with deviation much larger than expected winrate so results can be all over the place.

Yes, there are such things and knowing that my results are simply 1 in 10 occurrence over the past 450k hands is comforting to know.

However, that's not what I'm asking. Given an equity edge %, what's the probability of being above or below expectation for a given sample?

It might be really simple. GII with 70% equity on average and you'll run above EV, on average, 70% of the time. It's obvious in the single case but over many cases, I don't know. I have some stats knowledge but not enough to confirm to myself that it's a yes. Plus reality is not giving me any help on that score.

The likelihood that you are up trends to 100% as long as you have any edge at all, no matter how small. If his were not the case there would be no professional gamblers, and no casinos.

If you want to know the exact probability of being up over a finite sample, that’s doable, but not trivial. You need to find the sum of the probabilities of each positive result, which can be done using the binomial distribution. I don’t really have the motivation to do this for you right now, but I can tell you that with 70% equity it will take very few trials before you can be fairly confident that you’ll end up ahead.

Edit: Also, the numbers in the OP seem off. I’d double check them.

He doesn't want to know the chance of being up. He wants to know the chance that he will experience an outcome that is above expected EV, given a number of trials. As an example, with 1 trial you are above expected EV whenever you win, because your expected EV is a fraction of the pot, but you get the whole pot.

He's concerned because there doesn't seem to be a convergence in the number that he's looked at - it bounces all over the place. I haven't looked at his numbers to see if they're right, and I don't have a good intuition as to whether it "should" look like this or not. Just wanted to clarify what I think his actual question is.

I would consider constructing an example with easier numbers to make things clearer, personally. If I have some time I'll look at it later.

I think we've had a previous thread on this topic. It relates, as people above have said, to the skewness of the binomial distribution (how much of the distribution is above the EV).

I'll admit that I am not motivated enough to search for a previous thread or re-derive results that have previously been derived.

RustyBrooks - you are correct. That is my question.

whosnext - Thanks for mentioning skew. I've forgotten so much statistics that I couldn't search for an answer. I only remember asymptotically because it's one of the last terms I learnt despite seeing it everywhere for years and never being sure what it meant.

Intuitively it seems like asymptotic behavior should be an even distribution between above/below EV. Is this not the case?

Asymptotic implies large sample size. The Central Limit Theorem tells us that a "well-behaved" statistic approaches a symmetrical normal distribution as sample size increases. So, yes, an even distribution above and below EV is to be expected.

You just want the following. Given that I have a p probability of winning a single trial, what's the chance that, after n trials, I win more than n*p times (you run good if you win more than your fair share, which is n*p).

This can be calculated with the cumulative binomial distribution, which is implemented in many languages. In R for instance we can define the following function:

and see how it varies depending on the number of trials:
If you run the plot, you'll see that the probability keeps oscillating around 0.5 while approaching to it as n grows.