# Bayes’ Theorem in Plain English

## Simplest explanation of Bayes’ Theorem

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Assume we have a hypothetical machine learning model that has been used to obtain predicted class values (**class 0** and **class 1**) as shown in **Table 1** below.

Our system is a binary system with 2 classes, that is, **class 0** and **class 1, **with a total of 10 rows.

Let’s start by defining some basic probabilities (n = 10):

where

*P(Ye = 1)*is the probability that the**exact**value*Ye*is 1*P(Ye = 0)*is the probability that the**exact**value*Ye*is 0*P(Yp = 1)*is the probability that the**predicted**value*Yp*is 1*P(Yp = 0)*is the probability that the**predicted**value is*Yp is*0

## Bayes’ Theorem for Class 1

Now let us focus on the **class 1**, then from T**able 1** above, we define the following conditional probabilities:

where

*P(Yp = 1|Ye = 1)*is the conditional probability that the**predicted**value*Yp =1*given that the**exact**value*Ye = 1**P(Ye = 1|Yp = 1)*is the conditional probability that the**exact**value*Ye = 1*given that the**predicted**value*Yp = 1*

Putting these together, we have

Let *Ye = 1* be **event A** and *Yp = 1* be **event B**, then we can rewrite the equation above as

## Bayes’ Theorem for Class 0

For **class 0**, we can define the following conditional probabilities: