What is conditional probability?

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Conditional probability

Understanding math will make you a better engineer.

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Conditional probability is the single most important concept in statistics.

Why? Because without accounting for prior information, predictive models are useless.

Here is what conditional probability represents, and why it is essential.

Conditional probability

Conditional probability allows us to update our models given new observations. By definition, P(BA)P(B \mid A) describes the probability of an event BB, when AA has occurred.

Here is an example. Suppose that among 100100 people, 3030 have COVID.

Conditional probability

Based only on this information, if we inspect a random person, our best guess is a 30%30\% chance of them having COVID.

This is not good enough. We can take a look at more information.

What about people exhibiting a symptom, like a fever?

It turns out that among the 100100 people, 4040 have a fever.

Conditional probability

Let's say that a random person with a high fever walks in. How does our probabilistic model change?

We can leverage the prior information (fever) to get a more precise diagnosis than the random 30%30\%.

Conditional probability

By taking a more detailed look, we notice that 2424 people with a fever also got COVID.

Conditional probability

Thus, we can compute the conditional probability by focusing only on people with a fever.

Conditional probability

Using a similar logic, we get that if the patient has no fever, the probability of COVID drops to 10%10\%!

Quite a difference between our model with no prior information.

Conditional probability

Conditional probability restricts the event space, thus providing a more refined picture. This gives better models, leading to better decisions.

Conditional probability

(The numbers in this post are all made up. There are much more precise methods for actual COVID diagnosis.)

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