Roll die with waiting

Question

Consider an experiment of repeatedly throwing a fair die. Starting from the first throw, if a 6 comes up, the experiment ends immediately. Otherwise, you wait as many minutes as the number that came up before the next throw. What is the expected duration of the experiment, rounded to the nearest minute?

I only know that six throws are expected.

Answer 1

Let the expected duration of the experiment be $E$ minutes.

• $\frac16$ of the time, 6 is rolled, which stops the experiment immediately.
• $\frac16$ of the time, 1 is rolled, which makes you wait 1 minute and then roll again; the expected duration is now $1+E$ minutes. Similar reasoning applies for rolls of 2, 3, 4 and 5.

Thus we can make a relation for $E$: $$E=\frac16\cdot0+\frac16(1+E)+\frac16(2+E)+\frac16(3+E)+\frac16(4+E)+\frac16(5+E)$$ $$E=\frac16(15+5E)\qquad E=15$$ Thus the expected time of the experiment is exactly 15 minutes.