Monday, June 09, 2008

Probability of Disease - answer

A test of a disease presents a rate of 5% false positives. The disease strikes 1/1,000 of the population. People are tested at random, regardless of whether they are suspected of having the disease. A patient’s test is positive. What is the probability of the patient being stricken with the disease?

Most doctors answered 95%, simply taking into account the fact that the test has a 95% accuracy rate.

The answer is the conditional probability that the patient is sick and the test shows it - close to 2%. Less than one in five professionals got it right.

As for the answer. Assume there are no false negatives. Consider that out of 1,000 patients who are administered the test, one will be expected to be afflicted with the disease. Out of a population of the remaining 999 healthy patients, the test will identify about 50 to have the disease. (it is 95% accurate). The correct answer should be that the probability of being afflicted with the disease or someone selected at random from those with a positive test is:

= number of afflicted persons / number of true and false positives = 1/51

Alarming, isn’t it? If my doctor cannot interpret the test correctly, how can I trust him or her to treat me properly? I might actually do better to try to heal myself.

I found this case in “Fooled by Randomness” by Nassim Nicholas Taleb. An excellent book full of examples of the role of chance in life. It also tries to explain why we misinterpret probability. Fascinating and highly recommended.


Anonymous said...

I had a bad dream early this morning - I had to receive yet another surgery on my eye by a doctor referred to me by someone both of them were unknown to me (not the one whom I know well, who treated me succesfully last time). I refused to enter the operation theatre at the last minute and challenged the surgeon as to why we were so un-prepared - without telling me whether I needed to fast, my risk and possible consequences etc. It was horrible, really a nightmare.

I had bad experience in my first eye surgery by a doctor whom I think was behaving even less scientifically than me - I had to ask him to ascertain my recovery (3 weeks after the operation) with the optical scan rather than relying purely on his visual judgement and sadly I was proved right that the recovery / surgery was not successful as he presumed it would be! I would not recommend him to anyone! Ann

The Cat said...

While we're on the topic, here's more probabilities for you:

StephenC said...

Dear Ann, I am glad it was just a bad dream. The surgeries that you had earlier were bad enough. I pray that you won't have to go through anymore of that.

StephenC said...

Dear Cat, that NYTimes article is very interesting. I shall look up Mlodinow's book when I have time.

I find it fascinating that otherwise intelligent people often grossly misinterpret probability. Taleb's book provided some convincing reasons which match some of the things I learned from other sources recently (like how the brain works).