Standard probability round.
First head wins
The question
You and I alternate flipping a fair coin, you first. The first person to flip heads wins. What is your probability of winning?
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
2/3.
Let p be your winning probability. You win immediately with probability 1/2.
Otherwise, with probability 1/2 you flipped tails and the roles swap, so your opponent now faces the same position you did. Your chance from there is 1 - p.
So p = 1/2 + 1/2 times (1 - p), giving 3p/2 = 1 and p = 2/3.
The same recursion handles a biased coin: with heads probability q, p = q / (1 - (1-q)^2).
What it is testing
Setting up a self-referential equation instead of summing an infinite series by hand.
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