Before conducting an experiment, you need to have some information about the event you want to observe. In this case, the sample space changes and the event has its probability of occurrence changed.

## Conditional Probability Formula

** P (E_{1} and is_{2} and is_{3} and… and E_{n-1} and is_{no})** it's the same as

*P (E*.

_{1}) .P (E_{2}/AND_{1}) .P (E_{3}/AND_{1}and is_{2})… P (E_{no}/AND_{1}and is_{2}and is_{no}-1)Where :

- P (E_{2}/AND_{1}) is the probability of occurring E_{2}, conditioned by the fact that E_{1};

- P (E_{3}/AND_{1} and is_{2}) is the probability to occur AND_{3}, conditioned by the fact that E_{1} and is_{2};

- P (Pn / E_{1} and is_{2} and is_{no}-1) is the probability of occurring AND_{no}, conditioned on the fact that E_{1} and is_{2}… AND_{no}-1.

### Example:

One ballot box has 30 balls, 10 red and 20 blue. If a draw of 2 balls occurs, one at a time and without replacement, how likely is the first to be red and the second to be blue?

### Resolution:

Let the sample space be S = 30 balls, let's consider the following events:

A: red on first withdrawal and P (A) = 10/30

B: blue on second withdrawal and P (B) = 20/29

Like this:

P (A and B) = P (A). (B / A) = 10 / 30.20 / 29 = 20/87

Next: Independent Events