Imagine you were feeling unwell and went to the doctor to have a check-up. A blood test is carried out and the doctor tells you that you have a rare terminal disease which only affects 0.1% of the population. This blood test is 99% accurate and you start to panic. A test that has 99% accuracy means there is only a 1% chance that the result is wrong – this is bad news. However, a closer look at the probabilities will show that things are not as bad as they seem.

When you take the blood test, there are four possible scenarios:

**The blood test is positive and you have the disease**

The percentage of people who have the disease and will be correctly identified equals the test accuracy multiplied by the base probability = 99% * 0.1% = 0.099%

**The blood test is positive but you don’t have the disease – a misdiagnosis**

The percentage of people who don’t have the disease but will test positive will be = 99.9% * 99.9% = 98.9%

**The blood test is negative but you have the disease – a misdiagnosis**

The percentage of people who have the disease and will test negative will be = 1% * 0.1% = 0.001%

**The blood test is negative and you don’t have the disease**

The percentage of people who don’t have the disease and will test negative will be = 99.9% * 1% = 0.999%

The probability that you actually have the disease is therefore = 0.099% / (0.099% + 0.999%) = **9%**.

Even though the blood test is 99% accurate, the fact that the disease only affects 0.1% of the population drops the odds of you having the disease down to 9% – much better than 99% implied by one blood test alone.

The smart thing to do at this point would be to get another blood test at another clinic. This time, the prior probability of you having the disease is 9%. It has updated from the initial 0.1% because the first blood test was positive. If the second blood test is positive, the probability that you have the disease shoots up to **91%. **Two positive blood tests significantly increases the odds that you have the disease.

The difficulty is always estimating the prior probability. In this example, the prior probability that you had the disease was easy to figure out since only 0.1% of the population has the disease. In investing determining the prior probability is more difficult.

The point of Bayes theorem is not to be overly precise about probabilities – you always need to be wary of false precision. In investing, there is not much difference between a 90% or 70% probability of some outcome occurring in one year. It does matter in the long run because probabilities can compound.

Applying Bayes theorem can be useful in turnaround situations. If you initially assign a high probability to a new management’s ability to turnaround a company within a time frame that they set for themselves and they fail, you need to update your prior probability.

Similar to the blood test example, a small increase in the prior probability significantly changes the likely outcome. As Buffett said, “In the world of business bad news often surfaces serially: you see a cockroach in the kitchen and as the days go by you meet his relatives”.

Updating your prior probabilities is a good way of avoiding future cockroaches once you spot the first one.