Risk & Reward

Pricing a Die Roll

One roll of a fair die pays its face value in dollars, expected value 3.5. The fair ticket price is exactly 3.50 dollars, and anything above it is the seller's compensation for risk or margin.

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A game lets you roll a fair die once. The payoff is $1 for each dot on the upturned face. At what price should the ticket be set?

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#Expected payoff

Each face from 11 to 66 is equally likely and pays its own value, so the expected payoff is their average,

E[payoff]=1+2+3+4+5+66=216=72=3.5.(1)\E[\text{payoff}] = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = \frac{7}{2} = 3.5. \tag{1}
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face value paid
Every face from 1 to 6 pays its own value and is equally likely, so the expected payoff is their average, 3.5. That is the fair ticket price, the break-even point between seller and player.

#The fair price

A risk-neutral ticket price is exactly the expected payoff, $3.50. Charge more and the seller profits on average, charge less and the player does, and at $3.50 neither side holds an edge. Any premium above that is what the seller would add to cover risk or turn a profit.