Risk & Reward

Roll Until Not a One

Roll a fair die until something other than a one shows up, and you are paid that face value in dollars. The ones cost only time, so what is this game worth?

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You roll a fair die until a number other than a one appears. When such a number appears for the first time, I pay you the same number of dollars as there are dots on the upturned face, and the game ends. What is the expected payoff?

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#The ones are only delays

A one pays nothing and simply sends you to another roll, so the entire payoff is decided by the first face that is not a one. Conditioned on the roll not being a one, the six outcomes collapse to the five faces {2,3,4,5,6}\{2,3,4,5,6\}, each equally likely with probability 15\tfrac15. The number of ones beforehand never touches the prize.

Discard the ones and the first scoring roll is equally likely to be any of two through six. The expected payoff is their average, four dollars, sitting right on the middle face.

#Expected payoff

Averaging the five paying faces,

E[payoff]=2+3+4+5+65=205=4.(1)\E[\text{payoff}] = \frac{2 + 3 + 4 + 5 + 6}{5} = \frac{20}{5} = 4. \tag{1}

#Result

The expected payoff is 44 dollars. The rerolls on a one drag the game out but leave the prize untouched, so it is just the mean of the five winning faces.