Die roll, a six lets you add a second roll

Question

You roll a fair six-sided die. If it shows a 6, you roll again and add the second roll's value on top of the 6. Otherwise (1 through 5) you keep the first roll as is.

On average, what is the value you end up with, in dollars?

Accepted Answer

Two cases, blended by their probabilities. Case A: you did not roll a 6. This happens for faces 1 through 5, which is 5 of 6, probability 5/6. The value is just that first roll, averaging: \frac{1+2+3+4+5}{5} = \frac{15}{5} = 3. Case B: you rolled a 6. This happens with probability 1/6. Your value is the 6 plus a fresh die (which averages 3.5), so: 6 + 3.5 = 9.5. Blend by probability: EV = \frac{5}{6}(3) + \frac{1}{6}(9.5) = \frac{5(3) + 9.5}{6} = \frac{15 + 9.5}{6} = \frac{24.5}{6} = \frac{49}{12} \approx 4.08. The rare 6 pays off big (it becomes 9.5 on average), which more than makes up for…

Die roll, a six lets you add a second roll

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