True or false, I keep hearing statements like "Your dollar value of wins at gambling will approach your expected wins in the long-run"

Is this an accurate statement? Why or why not?

The ratio between the two will approach 1 in the infinite run.

I would say it is not accurate.

Your return on investment percentage (or ratio as the other poster said) approaches your expected rate of return as your number of investments (N) increases.

However, the raw total won/lost will vary randomly around expectation from its current point.

That statement is absolutely false.

The average absolute difference in dollars between results and expectation gets larger as trial sizes increase. The ratio of the difference to the expectation is what gets smaller. This is simply a result of the denominator growing faster than the numerator. It's a square root relationship, but BOTH get larger.

The law of large numbers applies to the ratio, not to the absolute dollar difference and not to "variance". Variance as a quantity gets larger as trials increase, by definition and in practice.

For more detailed explanations see the Probability forum, this has been discussed a hundred times.

Correct.


You mean the ratio of the difference to the total number of hands or to the total amount bet, not to the expectation.

Your win rate in bb/100 hands approaches your true EV in bb/100 hands (assuming that the later doesn't vary in a way that makes it impossible to define).


That would mean that they approach each other as the original statement said, and the original statement is false. EDIT: This is clearly not true, and 10000MegaWat's statement is OK and not the same as the original false statement.

Yes, to the total, thanks.

No, it won't. This is exactly the point being made in the title.

No, that wont. For ex., , yet x+1 and x don't approach each other.

Also, "Your win rate in bb/100 hands approaches your true EV in bb/100 hands" is a different formulation of what i said. The ratio between win rate and true EV is the same, whether you measure both in bb/100, just bb, or $.

Same story as above... You agree with me and then say that i'm wrong. I say a/b -> 1, you say |a-b|/a -> 0. Same ****.

The average absolute difference in dollars between the result and the expected result, gets larger as sample sizes increase. It grows at at a rate of the square root of the sample size increase. The ratio of the two does not approach any number, much less 1.

oh yes it does



edit: pic for the lazy.

Oops, that was dumb of me. You're statement was OK, and not the same as the original.

Does the ratio of the result to the expected result approach 1 for a series of 0ev coinflips?

I think its best to think about it as averages converging to expectations.

Like the probability that Xbar-X is greater than any specific small number converges to zero. The ratio version obviously has a problem if the denominator is zero.

Anyway, I'm happy everyone here gets the point. Xbar converges to X because the denominator of the average gets large at a faster rate than the numerator.

However, your gambling wins and losses in dollar terms really matter in the sense that future gambles are independent of the past. I hear people act like short term losses should be ignored. That's fine from an emotional standpoint because you don't want losses to take a psychological toll, but the way the point is explained often messes up the math.