The importance of conditional probability (why screening is not always a good idea)Jade Goody died from cervical cancer. This has caused some people to call for the screening age for cervical cancer to be lowered to the age of 20. A laudable goal, perhaps? Indeed, if we have tests, why don't we screen for every type of cancer?
Sadly, unless a test is very good, there will be many false positives- a simple example: Suppose 1 in a 1000 people gets cervical cancer, and lets suppose our test is 99% accurate. That is, it will miss the disease one percent of the time, and falsely claim the disease is present when it is not 1% of the time. Bayes theorem allows us to calculate the probability that someone who is declared positive for the disease actually has it. Rather than subject you to the equation, I will explain how it actually works.
Suppose I scan 1000 people. Of those, 1 of them will (on average), have the disease, so we correctly identify this with a 0.99 percent probability. There are 999 people remaining, and our test is 99% accurate, so thats 9.99 people diagnosed falsely with the disease.
So thats 0.99 who actually have the disease, and 9.99 who do not! And this example is actually generous: Generally speaking tests are MUCH worse than this, and I'm not sure the disease is even that prevalent.
Now one can increase these probabilities greatly by repeating the test, providing that we accept that our patient has the disease only if both are positive. Still, this is a lot of cost, and worry for the patient who has endured this. This is why screening tests are generally saved for those at risk to the disease. Bear in mind that we only have finite resource, and if the NHS spends a lot on screenings tests, while some people who wouldn't be picked up won't be, who knows who will suffer thanks to the massive waste in resources in checking these hundreds who do not actually have the disease.
This result is not immediately obvious, when looking at probabilities, but it is vital, and sadly not known by the majority of people