# Independent Events

We say that E1 and is2 and isn-1, ANDno they are independent events when the probability of one occurring does not depend on whether or not the others occurred.

## Independent Event Probability Formula:

P (E1 and is2 and is3 and… and Eno-1 and Eno) = P (E1) .P (E2) .p (E3)… P (Eno)

### Example:

One ballot box has 30 balls, 10 red and 20 blue. If we draw 2 balls, 1 at a time and replacing the draw in the ballot box, what is the probability of the first being red and the second being blue?

### Resolution:

Since events are independent, the probability of leaving red in the first withdrawal and blue in the second withdrawal is equal to the product of the probabilities of each condition, ie:
P (A and B) = P (A) .P (B).

However, the probability of being red in the first withdrawal is 10/30 and the probability of being blue in the second withdrawal 20/30. Hence, using the product rule, we get: 10 / 30.20 / 30 = 2/9.

Note that in the second withdrawal all balls were considered, as there was replacement. Thus, P (B / A) = P (B), because the fact that red ball came out in the first withdrawal did not influence the second withdrawal, since it was replaced in the ballot box.

Next: Event Merge Probability