Probability and dynamic-programming round.
The final roll
The question
You roll a fair six-sided die until the running total first exceeds 12. What is the probability the final roll is a 6?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
Approximately 27.93%.
Before the last roll, the running total must be one of 7 through 12. For each possible total, count the roll values that would cross 12 and note whether 6 is among them.
The clean route is to define the probability of eventually finishing on a 6 from each current total, then work backward from totals 12 to 0. The recurrence averages the next six states, with an immediate success when a 6 crosses the boundary and an immediate failure when another value crosses it.
Carrying that recurrence back to a starting total of zero gives 3,647,371,105 / 13,060,694,016, or approximately 27.93%. The stopping rule breaks the apparent symmetry between the six possible final rolls.
What it is testing
Whether you can turn a stopping-time question into a finite backward recursion.
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