Re: The Law of Large Numbers

Posted by Bisser on June 26, 1997 at 14:26:27: In Reply to: the riddle revealed posted by Dr. N on June 26, 1997 at 13:57:27:

Yes, there is something called Law of Large Numbers. These expression is used to refer to two asymptotic results known as the Strong Law of Large Numbers and the Weak Law of Large Numbers. Other similar results can be obtained by defining different modes of convergence on probability spaces (I believe one can formulate, for example, the Lp Law of Large Numbers using convergence in Lp norm, though I have never seen this done anywhere in the litrature). What you call the Central Limit Theorem above is in fact the Strong Law of Large Numbers. The Central Limit Theorem is a statement about asymptotic normality. In the above example, the Central Limit Theorem comes down to:

Let K be the number of sevens rolled in N rolls. Then as N gets large, the distribution of

(K/N - 1/6) / sqrt(5*N/36)

will converge to a normal(0,1) distribution.

Except for this terminology glitch your answer is correct. The first 400 rolls (or even the first 1,000,000 rolls) are irrelevant for asymptotic results as their effect is wiped out by the zillions of rolls which are needed for the "long run".

Bisser

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