Risk & Reward

Pricing a Die Roll

A game pays a dollar for every dot on a single roll of a fair die. What ticket price makes the game fair?

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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.