Sunday, July 15, 2012

A question of probability

In my post today, I'd like to ask readers of this blog a little question on probability that we* used a few years ago as an interview question for Oxford physics admissions.

The Oxford interview process is quite famous — or notorious, depending on your point of view — in the UK, and every year, come admissions time, newspapers run stories containing collections of "impossible" or eccentric questions that have supposedly been asked of candidates by malicious interviewers. In some newspapers the slant then given to the story is that the entire interview process is, by design or by coincidence, weighted against applicants from state schools, who are less well prepared for such shocks and are reduced to nervous wrecks.

I don't wish to enter into such a political discussion here, but in case any prospective interview candidates were to read this blog post I'd like to assure them that the interviews — at least in physics and at least in my experience — are certainly not like this. Interviewers realise that nervous applicants do not reflect their true potential, so they do not deliberately add to the tension and will in most cases attempt to create as relaxed an atmosphere as possible. They also realise that interviews, because of their highly subjective nature, are statistically not very good indicators of the true physics ability of the student. It is well-known that performance in the physics aptitude test applicants are required to take is a better predictor of their subsequent performance when at Oxford, so test results are substantially weighted up relative to interview scores when making decisions on who to admit. (You're also probably better off reading Oxford's own guide to sample interview questions than anything in the press!)

Anyway, I digress. The point of this post was the question of probability, not a discussion of admissions procedures. I thought the question was quite a nice one, but as it turned out no interviewees answered it correctly when it was first put to them, and after being given a gentle hint or two, almost all of them got to the right answer. So it was pretty useless at distinguishing between applicants and as a result not a very successful interview question. On top of that, I've also recently discovered that it is explained in great detail in a best-selling popular book on probability, so it probably won't be used again — which is why I feel it is safe to disclose it here! So here it is, slightly reworded:
You learn about the existence of some rare disease, X, which is known to affect 1 in every 10,000 people. Being a bit of a hypochondriac, you are afraid that you may have disease X, so you go to your doctor for a blood test. The doctor tells you that the test for this disease is accurate 99.9% of the time. To your horror, the test result comes back positive. What is the probability you have disease X?
I won't give the answer immediately (update: the answer is now explained here), because I'd first really like to know how readers of this blog would answer it. So I'd like to encourage everyone who reads this post to vote in the following poll -

What is the likelihood you have disease X?

Please don't be shy about voting! If you get it wrong, well, you've done no worse than some of the brightest A-level students in the UK. On the other hand, you could have the satisfaction of getting it right and displaying your knowledge to the world (the poll's completely anonymous, but still). Feel free also to comment on this post, but please don't give the answer away if you know it.

The reason that I've posted this question and the reason I'd like to know people's answers is that although it wasn't a good interview question, I think it is still a good way to get people to think about probability, and the question of how to deduce information from limited evidence. These are topics which I would like to discuss in a series of further posts over the next few weeks, so I'll use this as a starting point.

* I took part in admissions interviews between 2007 and 2010 as part of a tutoring job while finishing my doctorate. This particular question was one suggested by my (senior) colleague for use in one of those years.


  1. I went with my gut instinct; I can see that there is already a fair spread of answers, even among the first ten respondents...

  2. Ooh I seem to have gone with the most popular option at the moment...Do let us know what the correct answer is!

    1. Hello Anonymous. I certainly will, but I'll let a few more people have a go first. I'll put the answer and explanation into a separate post in a few days' time.