Dice Expected Value with One Reroll: The Optimal Strategy
Trading floor interview style (expected value)
The riddle
You roll a fair six-sided die and win, in euros, the value of the face you get. You may roll the die once more, if you wish; the second result is then final. What is the expected winnings with the best strategy?
Need a hint?
First calculate the average of one roll, then compare it with the faces you would keep after a reroll.
Show the answer and the solution
Answer: €4.25: reroll if the first result is 1, 2 or 3, and keep the first result if it is 4, 5 or 6.
The reasoning
Without a reroll, the expected value is the average of the six faces, that is (1 + 2 + 3 + 4 + 5 + 6) / 6 = €3.50. With one reroll, the best strategy is to reroll if the first result is 1, 2 or 3, since €3.50 is higher than each of those faces. If the first result is 4, 5 or 6, keep it. When you keep it, the gain averages €5 with probability 1/2; when you reroll, it averages €3.50 with probability 1/2. The expected value is therefore 1/2 × 5 + 1/2 × 3.50 = 2.50 + 1.75 = €4.25.
What next?
Reasoning improves like memory: by practising often. Next, test yourself on our general knowledge quizzes.