Two fair coins are flipped simultaneously. This is done repeatedly until at least one of the coins comes up heads, at which point the process stops. What is the probability that both coins came up heads on the last flip?
Source: NCTM Mathematics Teacher 2006

SOLUTION
To get a feel for the problem we conduct $120$ trials using the random generator function of a TI-83 Plus. The results are recorded in the spreadsheet below

While some trials end quickly with one double flip, others took longer with several double flips. What is important is not how long the trials take but how they end. They end in $3$ possible ways: HH, TH, or HT. The probability that both coins came up heads on the last flip equals $1/3$.
The spreadsheet shows that in the $120$ trials the event HH occurs $40$ times. The experimental probability = $40/120=1/3$ dovetails with the theoretical probability.

Answer: $1/3$