If you are a flower then April is allegedly the cruellest month, but if you are a student of any sort then I'm sure you would have picked June. June is a month of exams, exams and more exams for school and college students everywhere. One of the more common forms of exam paper that is devised to aid quick marking is the *multiple choice* question paper. You have to pick the right answer from a suite of alternatives.

If it all gets too confusing, should you try guessing?

My most memorable encounter with this format was when I first saw the written part of an American driving test. It asked what you should do if you started to feel drowsy whilst driving on the freeway. The options were (a) speed up so as to get to your destination sooner; (b) do an emergency stop; or (c) wait until you arrive at a place where it is safe to stop, then pull in and rest until refreshed before continuing safely on your journey.

Other tests can be more demanding though. One of the features of all multiple choice tests is that they need to be as immune as possible to random guessing. You don't want to give a passing mark to a candidate who just guesses at random. One way to discourage a guessing strategy is to penalise wrong or blank answers with negative marks. Let's see how we might do that.

Suppose that there are five answers to choose from and we give one mark for the correct answer but subtract marks for any of the four wrong answers, or if no answer is chosen. If you guess answers at random then there is a 1 in 5 chance that you will choose any one of them and your expected average score on each question will be

that is, a 1 in 5 chance of guessing the right answer and scoring 1, and a 4 in 5 chance of guessing wrongly and scoring Ideally, you might want the reward for guessing to be zero. This will be achieved if you make the penalty mark equal to More generally, it is easy to check for yourself by the same argument that if we had answers to choose from then the expected score from a strategy of random guessing will be zero if we choose a penalty of

We can also work out the variance, which measures by how much the score of a question differs from the zero score, on average, if you're guessing the answer. Our simple statistical process gives an outcome of with probability and an outcome of with probability The variance is the average value of the square of the outcome minus the square of the average, which we have designed to be zero — so it's just the average value of the square of the outcome. This makes the variance in the score per question

### About the author

John D. Barrow is Professor of Mathematical Sciences at the University of Cambridge, author of many popular science books and director of the Millennium Mathematics Project of which *Plus* is a part.