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Say we are playing a casino game where we have there potential outcomes, and 3 probabilities associated for each one, all which sum to 1.

They are:

  1. Win $1.00 (80%)

  2. Lose $4.00 (17%)

  3. Lose $1.00 (3%)

This gives us an expected value of (.8*1) + (-4*.17) + (-1*.03) = $0.09

Now we know the expected value, but I am wondering how would you then quantify the edge to a single probability, like 2% or 3%, in this scenario?

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You would typically divide the the value of the expected outcome by the value of the gambler's original stake (to deal with issues in blackjack like doubling down or splitting).

Here the stake looks to me like being $\$4$ since that is the amount most commonly lost, so I would have thought it is reasonable to say that the advantage here is $\dfrac{0.09}{4.00} = 0.0225$ i.e. $2.25\%$

On the other hand, if your quoted gains and losses are from the house's perspective where the gambler is betting $\$1$ and the gambler is expected to lose on average, then I would say the house advantage was $\dfrac{0.09}{1.00} = 0.09$ i.e. $9\%$

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