Here is a part of Bob Anderson's answer in my question (Does variance do any good to gambling game makers?)
Suppose you had two gamblers who were flipping coins against one another with fair odds for 1 dollars a flip. One of the gamblers has 20 dollars and the other has 100 dollars. If someone goes broke the game is over.
what is the formal mathematical formula/distribution that link "the variance and the odds that the gambler who has smaller budget would win" together and what is its proof?