One little expansion: what you described first is the arithmetic expectation, which is a good approximation when the numbers involved are small compared to tota

One little expansion: what you described first is the arithmetic expectation, which is a good approximation when the numbers involved are small compared to total wealth. When you start taking larger bets, you want the geometric expectation of total wealth, i.e. (current wealth + outcome)^probability. (This is the Kelly criterion for judging significant opportunities.)