Originally Posted by
JohnGalt007
Hi PeterLee,
I read through the paper as well as the proof of the theorem you mentioned several times. I paid particular attention to the mapping argument used by Thorp to justify the invariance theorem. Here is my takeaway insofar as it relates to your confusion about why his mapping argument works:
Imagine I have a biased D6 die with probability distribution P(X=1) =/= ... =/= P(X=6). It is not strictly necessary that the probabilities are all distinct, but it does allow us to follow the logic more closely. The expected value of the number rolled on the upppermost face can easily be computed as E(X) = 1*P(X=1) + ... + 6*P(X=6). Permute the listing of the values of your random variable in any one of 6! ways. For example, instead of listing X=1,2,3,4,5,6 and P(X=x) = 1/21,2/21,3/21,4/21,5/21,6/21 in that order, write X=5,2,3,1,6,4 and P(X=x) = 5/21,2/21,3/21,1/21,6/21,4/21. This can be done either at random or according to a deterministic decision criterion; either way, there exists some rule that maps the ordered listings of values and attendant probabilities with the rearranged listings. Call this rule f. f is guaranteed to be one-to-one and onto, so that every value of my random variable along with its probability of occurrence occurs once and only once post-arrangement. Thus, after rearranging everything in the listing of my random variable I'm still able to write out the probability mass function and compute the correct expected value, even if it may be slightly inconvenient for me to list out everything now. Using my example earlier, E(X) = 5*5/21+2*2/21+...+4*4/21 is still the correct expected value E(X) = 1*1/21+2*2/21+...+6*6/21 thanks to the commutative property of addition of real numbers. Again, the crux of the argument is that the conclusion only works because the rule f by which I rearrange the listings of values and probabilities of my random variable X is one-to-one and onto, so nothing essential is lost and nothing inessential is gained post-arrangement.
Thorp applied similar reasoning to the discrete random variable representing the player's net profit in this game where cards are dealt without replacement instead of this die roll where dice are rolled with replacement, and he had to address different cases depending on which cards were dealt initially. The underlying mapping argument still remains valid. Hope this helps!
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