Conditional Probability for Normal People

This important concept is usually taught with horrific formulas — here’s an intuitive explanation instead.

Conditional probability lets you answer important questions like: given that I tested positive for malaria, how likely is it that I actually have malaria? So, as you can imagine, it’s pretty important that doctors and nurses understand how it works.

The problem is, most schools teach conditional probability (also known as Bayes) in an incoherent way with a terrible formula. So, doctors memorize it for a statistics class, then promptly forget it. There’s a better way that lets you avoid the formulas altogether and get the same results.

Here’s an example.¹

Let’s assume:

• About 1% of people have malaria, worldwide.

• We have an extremely accurate test for malaria.

• Anyone who has malaria, and takes the test, will be described as sick. We do not have any cases of people who have malaria but the test says they are healthy.

• Of people who do not have malaria and take the test, 1% of them will be described as sick even though they are actually healthy.

The question is:

• I just took a malaria test, and it said that I am sick. What are the odds that I actually am sick?

Intuitively, I would guess that the odds are somewhere around 1%. Actually, it’s closer to 50%. Here’s how you get there.

Let’s make up an example with specific numbers — the trick is to actually do the math.

Here’s the breakdown for a group of 1 million people:

For this example, I’ve gone through and plugged in the percentages and the numbers. So, since 1% of the population as a whole have malaria, and we have 1 million people, 1% of 1 million is 10,000 people have malaria. I picked one million because it’s a nice round number that makes the calculations easier.

OK, my question is: I tested positive for malaria, so what are the odds that I am actually sick? Well, you need two numbers for this:

• the number of people who actually have malaria

• the number of people who tested sick for malaria — both the ones who really are sick and those who are healthy but tested sick

Let’s calculate those two numbers:

• the people who actually have malaria is 10,000

• the people who test sick is the sick people who tested sick + healthy people who tested sick = 19,900

Out of all the people who tested sick, what percentage of malaria?

• 10,000 / 19,900 = 50.3%

I only have a 50% chance of actually being sick!

So, why do we get such strange results? It’s because the overall percentage of people who have malaria is so low. Because only 1% of the population has malaria, the number on the top of our fraction (10,000) is also low. You get much more intuitive seeming results if you look at a more common disease. Here are the numbers for an illness that half the population has:

In this case, if I test sick, the odds I really am sick are a lot more intuitive:

• total sick people / total people who tested sick

• 500,000 / 505,000 = 99%

Because the number of sick people is much higher then with malaria, i they make up a much higher percentage of the people who tested sick who are actually sick.

Congratulations — you now understand test results better then most doctors do! Go forth and enjoy.

¹ These facts about malaria are not accurate — we’re simplifying to make the logic easier to follow.

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