the riddle revealed

Posted by Dr. N on June 26, 1997 at 13:57:27: In Reply to: Law of Independent Trials: Is It Flawed? posted by NoMath on June 26, 1997 at 12:44:05:

Your question pertained to an interpretation as to what is meant when some refer to the so called "law of large numbers".

"If say after 400 rolls of a pair of dice, a seven has not yet been observed....since in the long run 1/6 of the rolls should be sevens...we must now have a probability of more than 1/6 to eventually catch up"

My reaction:

First I might note that in math texts, their really is no such thing as literally the law of large numbers. There is an important theorem called the 'central limit theorem' that is related - but often misinterpreted.

In simple terms, lets first define p as the probability (or proportion of the time) we observe a roll of seven. Then what the law of large numbers states is that as the number of rolls approaches infinity - the observed proportion of sevens will approach the value of p=1/6. The formal definition of what I mean by "approach" can be placed in mathematical terms. But it DOES NOT mean that literally 1/6 of the number of tosses must be sevens.

Think of it this way. If we imagined billions and billions of tosses, what effect does our prior info of 400 tosses without a seven have. At first, it may seem that we are owed '66.67' more sevens than usual. But when we imagine computing an actual value of the 'observed propoertion' of sevens, we count all the sevens observed after billions of tosses and divide by this very large number. The seemingly large effect of what we thought to be 66.67 owed tosses, actually only shows up as a difference of say .0000000001 from the theoretical value of p=1/6.

I'm not sure if this will help, or if I've just made it seem more confusing? The notion of difficulty is in expressing what it means for an observed frequency to 'converge' to a theoretical value (probability).

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