A die is rolled repeatedly until two consecutive rolls have

A die is rolled repeatedly until two consecutive rolls have the same number. How many times do you expect to roll the die?

Solution

Let E be the expected number of rolls.

Suppose, we expect the number 6 to come twice. Assume that there are 6 rolls until a 6 comes. Now, there is a 1/6 chance to roll once more and 5/6 chance of starting all over again.

Thus, E=6+1/6*1+5/6(E+1)

E=42