Understanding expected value
Multiply each possible outcome by its probability, then add the products. If a game has only one prize and otherwise nothing, that sum is just probability × prize, because the "nothing" outcome adds 0. Comparing two offers that cost the same to play (here, nothing), the one with the higher expected value pays more on average. Offer A: 3/5 chance at 100 → EV = 60. Offer B: 3/10 chance at 250 → EV = 75. B pays more on average, though it wins less often. If playing costs something, subtract the cost before comparing.
Seeing it in action
Worked example
A free game: 3/5 chance to win 100 points, otherwise 0. Expected value?
EV = 3/5 × 100 + 2/5 × 0 = 60 points.
Expected value = probability × payoff.
Worked example 2
A spinner pays 10 points on 1/2 of spins, 40 points on 1/4, and 0 on the rest. Expected value?
EV = 1/2 × 10 + 1/4 × 40 + 1/4 × 0 = 5 + 10 + 0 = 15 points.
A common mistake, and the fix
Averaging the results without their chances.
For the spinner, (10 + 40 + 0) ÷ 3 ≈ 16.7 treats the three results as equally likely, but they aren’t. Weight each result by its chance and add: 10 × 1/2 + 40 × 1/4 + 0 × 1/4 = 5 + 10 + 0 = 15 points.
Try a few
1/4 chance at 80 — EV?
1/2 chance at 50 — EV?
Which is better: 1/3 of 90, or 1/2 of 50?
Try it in Numeris.
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