# Question

What does Bayes’ Theorem mean? What do the different parts of the equation say in plain language?

Let’s first recall that Bayes’ theorem is stated as follows

`P(A|B) = P(B|A) * P(A) / P(B)`

In plain language, Bayes’ theorem is saying

If we know that B is true and would like to find when A is also true, we should

• focus our attention on the outcomes where B is true
• and then once focused there, look at all the outcomes in this area where A is true.

The first point first. Since all probabilities are between 0 and 1, if we consider the number 1 to mean all possible outcomes, then `P(B)` means all outcomes where B is true. So this is where the denominator of `P(B)` in the equation comes from.

For the second point, we’re asking here about the intersection of A and B which we can write as `P(B|A) * P(A)`, explaining the numerator of the equation.

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Suppose you know the probability of A (and consequently of not A), and for each of these outcomes you know the probability of B/not B.

You want actually to reverse the prediction and figure out the proba of A if B subsequently happens. This corresponds to a fraction of the ensemble where B happens, because this ensemble also includes the event that A doesn’t happen and B does.

Consequently, the proba of this event you look for is equal to the probability of the event that A happens and then B divided by the overall probability of B happening.

This can be depicted by the scheme below

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