2025 SSMO Team Round Problems/Problem 5

Problem

Bob rolls a fair six-sided die until he rolls an even number. What is the expected sum of all the numbers he rolled?

Solution

Let $S$ denote the expected sum of all the numbers Bob rolls. If Bob's first roll is an odd number $n_o$ , his expected sum is $n_o + E$ . If Bob's first roll is an even number $n_e$ , his expected sum is simply $n_e$ . Therefore, $E = \tfrac{1}{6}((E+1)+(E+3)+(E+5)) + \tfrac{1}{6}(2+4+6)$ . Solving, we find $E = \boxed{7}$ .

~Sedro

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2025 SSMO Team Round Problems/Problem 5

Problem

Solution