Expected Value of Rolling a Die With Rerolls

The optimal one-reroll strategy keeps a roll of four or higher and rerolls anything lower, producing an expected value of 4.25.

A fair six sided die is rolled once, and the player may choose to reroll it exactly one time before the final value is locked in. Under optimal play the expected value of the final result is 4.25, achieved by keeping any roll of four, five or six and rerolling on one, two or three. The threshold comes directly from comparing the value already showing against the expected value of a fresh roll, which is 3.5.

The setup has one die, one roll, and one optional reroll. The player sees the outcome of the first roll before deciding whether to keep it or replace it with a second, final roll. There is no third chance, and the decision has to use only the number currently showing. This is the simplest member of a broader family of optimal stopping problems, where a decision maker sees outcomes one at a time and must choose when to stop.

The die is fair, so each face from one through six comes up with probability one in six on any single roll. Before any roll happens, the expected value of one die roll is the average of the six faces, one plus two plus three plus four plus five plus six divided by six, which equals 3.5. That number turns out to be exactly the threshold the whole problem hinges on.

The reroll only replaces the current value with a fresh roll, so the value of rerolling is always exactly 3.5 in expectation, independent of what the first roll showed. A rational player keeps the current roll only when it beats 3.5, and rerolls otherwise. Since die faces are whole numbers, keeping a roll of four or higher and rerolling one through three is the unique optimal rule, since no face equals 3.5 to create a genuine tie.

To compute the expected value under this rule, split the six equally likely first roll outcomes into two groups. Rolls of four, five and six are kept as they land, contributing four plus five plus six, or fifteen, weighted by one sixth each. Rolls of one, two and three are always rerolled, and each of those three outcomes leads to the same 3.5 expected value from the second roll.