Solution
Six Different Faces
Show the problem again
Six fair dice are thrown together. If the six faces shown are all different (so that 1, 2, 3, 4, 5, 6 each appear exactly once), you stop; otherwise you throw all six dice again. What is the expected number of throws? Give an exact value.
Worked solution
The answer is 324/5 = 64.8 throws. A single throw shows six different faces with probability 6!/6⁶ = 720/46656 = 5/324. Throws are independent, so the number of throws is geometric with this success probability, and its mean is the reciprocal, 324/5. The surprisingly small success probability (about 1.5%) is the same arithmetic as the birthday problem: six people with six possible birthdays almost always share one.
Source: Classic geometric-waiting-time exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.