Opening question in trading interviews.
Roll a die, take it or reroll
The question
You roll a fair six-sided die and may either keep the value or reroll once and keep the second value. What is the optimal strategy and its expected value?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
Reroll on 1, 2 or 3. Expected value 4.25.
A reroll is worth the plain expectation of a die, 3.5, so keep anything above 3.5 and reroll anything below.
That means keep 4, 5 or 6 and reroll 1, 2 or 3.
Expected value is the average of 3.5, 3.5, 3.5, 4, 5 and 6, which is 25.5/6 = 4.25.
The natural follow-up is two rerolls. The same logic applies recursively: the value of having rerolls left, 4.25, becomes the new threshold, so you keep 5 or 6 and the answer rises to about 4.67.
What it is testing
Optimal stopping in its smallest possible form.
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