Another way of looking at this (actually just a rephrasing of what you've said) is that the mathematical expectation of a strategy is the sum of (the profit of

Another way of looking at this (actually just a rephrasing of what you've said) is that the mathematical expectation of a strategy is the sum of (the profit of each event multiplied by the probability of it happening.) That is expec = sum(prob(x)x) For an $ investment, $expect = sum(prob(x)x - invest(x)) Say we have two strategies: One invests nothing and is right 90% of the time, but pays 0 when it's right; the other invests $50,000 and is right 10% of the time. When it is right it pays 10x, or $500,000. exp(strategy 1) = prob(x)x - 0x = (0.9)0 - 0 = 0 exp(strategy 2) = prob(x)x - x = (0.1)10 -1 = 0 So if we only have a 10% success rate and a 10x payoff, we might as well be sipping tequila on the beach. But if we are smart, or hard working, and increase our probability to 20%: exp(strategy 2) = prob(x)x - x = (0.2)10 -1 = 1 which is a positive expectation, which is a good thing. Increasing the probability OR increasing the payoff increases the expectation.