Originally Posted by random_person
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?
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.