KV,

You attempt mathematical modelling of various gambling strategies. So it is disturbing that you cannot figure that Crash's formula is actually a simple version of Expectation.

I consider that Expectation is second to no other concept in risk/reward evaluation.

While I knew about the originator of Expectation, until now I didn't realise that he introduced it in the first ever published book on probability.

Here is that original 1657 work conveniently translated from the Latin:

http://math.dartmouth.edu/%7Edoyle/...ens/huygens.pdf

And here is a rework and attempted explanation is modern language.

http://www.math.dartmouth.edu/~doyl...hedge/hedge.pdf

While life is too short to more than browse through what most people should already kinda' know, it does give an insight in how people started to think about probability.

And it bears an eerie resemblance to the concepts required in Exchange Betting.