bayes theorem by the expert tutors at The Tutor Team


The Counter-intuitive results of BAYES’ THEOREM


bayes theorem by the expert tutors at The Tutor Team

No need to look up the meaning of these symbols, we just give you (one version) of Bayes’ Formula for reference.


All you need is explained below!




Especially relevant during a pandemic, is the question:

“I have been diagnosed with a disease, what are the chances I actually have the disease?”

 The reason that this question is even worth asking is that tests of many kinds, particularly medical, are not full-proof.  Test negative, you may still have the disease: test positive, you may be fortunate enough to not actually have the disease.  It all comes down to the reliability of the test and how Bayes’ Theorem comes into play.




You may well ask: why aren’t these tests properly reliable?

A sensible question, but the reasons are shall we say, technical and beyond the scope of this blog.


The easy-to-follow logic pans out as follows:

Imagine 500 people in 10 000 have a disease (5%) for which they are all tested.

Suppose the test is 95% accurate.

Even if we said ‘99% or 99.9% accurate’ the argument still holds perfectly true.

Of the 500 with the disease, around 95%, that is 475 will test positive, as they should.. after all, they have the disease.

However, that means 25 will test negative even though they do have the disease.

Now it gets interesting, because of the other 9500 who are disease-free, 95% or 9025 will test negative.

Hence, therefore, the other 5% which equals 475 will test positive, even though they aren’t.


If your result comes back labelled ‘positive’, then in point of fact your chance of actually being positive is 50%, not 95%, a decidedly cheerier figure owed entirely to the false-positive results which necessarily occur when test results aren’t 100% accurate.



Bayes’ Theorem is a simple example of how Maths can produce astounding predictions, but if you really want to see how massively counter-intuitive probability theory can be and drive yourself mad in a genuinely fun but devilish way, look up “The Monty Hall Problem”.

It’s that, more than anything that got me hooked on statistics to the point of obsession!


A bit about the author, Mark F:

Dr Mark F

Mark began tutoring around 1990, before which he taught in both state and private sectors since 1986; examples being Surrey College (Guildford), Horsell High School (Woking) and Farnborough College of Technology, and most recently at Seaford College in Petworth.

With a vast library of past examination papers in all his subjects at GCSE and ‘A’ level for all major exam boards: Edexcel, AQA, WJEC and OCR, Mark is a go-to tutor for Science!

You can enquire about tutoring with Mark here

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