# Multiplication Theorem on Probability and proof

#### alfred

##### New member
Hi Everyone!

I'm with Conditional Probability and I don't understan this theorem.

Theorem:
If

then

Proof:
All the conditional probabilities are well defined, since

We can rewrite the right site of the equality as follows

Obviously we can simplify the terms through

Can anyone say me how does the simplification work? And why it is so important to be sure that

Thank you =)

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Welcome to MHB, alfred!

The simplification is a consequence of how fractions are multiplied and simplified in general.
Consider for instance:
$$\frac 3 4 \cdot \frac 4 5 \cdot \frac 5 6 = \frac 3 {\cancel 4} \cdot \frac {\cancel 4} {\cancel 5} \cdot \frac {\cancel 5} 6 = \frac 3 6$$

#### alfred

##### New member
Thanks! You were right