Please explain how the following two statements are not mutually exclusive.

1. If a person runs lucky or unlucky over a sample of hands (actual winnings are greater or less than than expected winnings), that will have no relevance to how that person will expect to run in their next sample of hands.

I believe this is one way to express the concept of statistical independence, where the outcome of one event has no bearing on the outcome of a subsequent event.

2. In the very long run a person can expect that their actual winnings will be the same as their expected winnings.

I believe this follows from the definition of expectation; if you normalize a series of hands all in before the river to the ev of each hand then extrapolate to a huge sample size, the actual winnings will converge to the ev winnings.


I will describe a situation which seems to make 1 & 2 mutually exclusive.. any clarification is much appreciated.

A person has been very unlucky for their first sample of hands played. At this point their actual winnings are much less than expectation over this first sample.
Now we examine the same person's graph after playing millions of hands and we see that their actual winnings is indeed very close to their expectation.
After running unlucky in that first stretch of hands and also seeing their actual on top of their ev in the lifetime graph, this player concludes that after the initial run bad, they are going to have to expect to be running above ev for some sample in order to eventually get actual winnings to catch up to ev winnings.

This type of thinking obviously violates 1, but seems necessary if 2 is going to be true.

I may be erring by using vague terms like "a sample of hands", "very long run", or with "converge", but I am hoping some of you will understand the point I am trying to make and can articulate what I am missing in my understanding of variance.

Number 2 is not correct the way you stated it. The expected difference between actual winnings and expected winnings actually grows over time (at a rate proportional to sqrt(number of hands)). But your actual winnings grow much faster at a rate proportional to number of hands.

To rephrase it in a correct manner: In the very long run a person can expect that the difference between actual and expected winnings will become insignificant compared to their actual winnings (unless of course that person is an exactly breakeven player).

Or another way to put it: In the long run, a person's actual win-rate will converge to their expected win-rate. Not total winnings.

Yeah, this is the important thing.

Thank you for the replies, and bear with me.. I haven't quite resolved my confusion but I am confident this discussion will get me there.

Let's assume the person is a winning player (say ev 5bb/100 for example) and all hands played (1MM for example) are at .5/$1 blinds.

Since a win-rate is simply the number of blinds they expect to win in 100 hands, what is the mistake in saying their actual winnings will be converging to the ev of $50,000 for the described player ?

If we simplify this issue to flipping coins.. let's say we flip a coin 100 times and the first 40 are tails. If we calculate the expected number of heads in the 100 sample test to be 50 heads, why can we not expect to get many more heads in the next 60 flips, if we are going to get closer to the expected 50 heads vs 50 tails ?
Based on the posted responses, you might say our flipping will converge to .5 heads per flip, but we cannot say it will converge to 50 heads out of the 100 flips???

I may be slightly unclear about the exact long term prognosis for difference in ev and actual, but is there something fundamentally incorrect about the non-specific, laymens description that your actual winning's line will get closer to your ev graph as time goes on ?

Unless my understandings are still way off the problem described in OP seems to still exist... how can we not expect to run better in future trials if we are ever going to "catch up" to our ev line, or to the 50% likelihood in flipping coins?

Let's say you flipped a coin 100 times and got 60 heads. Your expectation was 50, so you ran ahead for that period. Your expectation for the future is still 50%. You expect 50 heads out of the next 100. So the expectation is that for the whole 200 flips, you'd have 110/200. 110/200 is closer to 50% than 60/100.

That's what is meant by saying you expect the average to converge to 50%.

But it is not the case that your expectation, at the end of 100 more heads, would be to hit 100/200.

If you don't believe me, try it. Flip a coin until you get 3 heads in a row. This will happen 1/8 times. Then flip 1 more time and record the outcome. Then start again, flipping until you hit 3 in a row, and so forth. If you do this enough you'll find that your 4th flip happens 50% heads and 50% tails, and that there is no bias towards flipping a tail even though you flipped 3 heads in a row.

Thanks again.. I can't disagree with anything you have said, but I am still encountering the same problem when I am thinking about this.

From what I understand in your example, our initial run good in heads is irrelevant because in continued trials, the expectation is always 50%, therefore our running average will decrease from the 60% it was after the first 100 flips towards the 50% we both know is the probability of heads on independent flips.
So once the first 100 flips have taken place with 60% heads, we say the ev of 50/100 flips for the next 100 makes the average for the ACTUAL first 100 and the EXPECTED second 100 flips to be 110/200 (55%). If we add a third expected trial on top of our first actual trial and second expected trial, we would say 160/300 (53.33%) heads brings our running average closer to the true 50%.

Still, unless we can be guaranteed that certain future 100 flip samples will have more tails than heads, we will never have <50% heads averages for those 100 flip samples that would be necessary to bring the initial 60% average down to the true 50% average it should be.

Obviously still some confusion with my understanding, but I do feel I am getting closer

Well, first of all, that's basically what "converge" means - it means to get closer with time or samples or whatever. There is no guarantee that it will actually *get* to 50%. But for example if you did 1 million more flips, with an expectation of 500k heads, then you'd be at (500,000+60)/(100+1000000) which is really really close.