Oxford Maths Interview: A Coin-Game Probability

Adriaan (Harry L.) · Mathematics (Oxford)

Hi, I'm Adriaan, and I study Maths at Oxford. I want to run through a short but instructive maths interview question about probability and expectation, using a simple coin game. It's a nice example because the setup sounds easy, and the real work is in spotting the structure hidden inside it.

The game

You toss a coin over and over. Every time it comes up heads, you win one pound and carry on playing. As soon as it comes up tails, the game ends and you keep whatever you've won so far. The question is: what is the expectation, the average amount of money you'd expect to win from this game?

Setting up the expectation

Let the probability of heads be h, so the probability of tails is one minus h. To win a given amount, you need a run of heads followed by a tail. The chance of winning exactly one pound is the chance of one head and then a tail; two pounds is two heads and then a tail; and so on. Writing the expectation as each possible payout multiplied by its probability gives you a sum whose terms look like one times a probability, plus two times a probability, plus three times a probability, and so on. Getting this setup down cleanly is half the work, and it's the half a lot of people rush past.

Spotting the series

This is an infinite sum, and the neat move is to recognise its shape. The series 1 + 2h + 3h squared + 4h cubed, and so on, is the derivative of a geometric series. We already know that the geometric series 1 + h + h squared + ... sums to 1 divided by (1 minus h). So instead of trying to add infinitely many terms directly, you can differentiate that closed form with respect to h to evaluate the sum. Recognising that a sum is the derivative of something you already know is the real insight the question is testing.

The result and a sanity check

Carrying the differentiation through and simplifying, the expected winnings come out to h divided by (1 minus h), the ratio of the probability of heads to the probability of tails. It's always worth sanity-checking a result. With a fair coin, h is one half, so h over (1 minus h) is one half over one half, which is 1. In other words, with a fair coin, the average amount you'd win from this game is one pound, which feels about right.

How it would actually run in the room

I've given the intended method fairly directly here, but in a real interview you'd have far more time to think, and you'd very likely get hints along the way. A question like this might take around twenty minutes, and the interviewer would expect to watch you set up the expectation, wrestle with the infinite sum, and gradually find the calculus connection, rather than producing the answer instantly. The process is the point.

What the interviewers are looking for

A few things earn credit here:

• translating the words into a clean mathematical setup before doing anything clever
• looking for structure in an infinite sum rather than trying to brute-force it
• connecting the sum to a series you already know, via differentiation
• sanity-checking the final expression against a simple case, like a fair coin
• making each step explicit so the interviewer can follow and help

Common mistakes to avoid

The usual slips are diving into algebra before defining the game and the variables clearly, panicking at the sight of an infinite sum instead of looking for a familiar structure, trying to add the series term by term, and skipping the sanity check that would have caught an error.

Final thoughts

Get quick at turning worded problems into a clean setup, get comfortable with geometric series and with differentiating and integrating them, and always test your final answer on an easy case. Those habits carry across almost every maths interview question, not just this one. If you'd like help preparing for Oxford maths interviews, Oxbridge Solution can help.