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Addition Rule in Probability

If $A$ and $B$ are two events in a probability experiment, then the probability that either one of the events will occur is:

$P (A or B) = P (A) + P (B) - P (A and B)$

This can be represented in a Venn diagram as:

$P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Math diagram

If $A$ and $B$ are two mutually exclusive events , $P (A \cap B) = 0$ . Then the probability that either one of the events will occur is: $P (A or B) = P (A) + P (B)$

This can be represented in a Venn diagram as:

$P (A \cup B) = P (A) + P (B)$

Math diagram

Example:

If you take out a single card from a regular pack of cards, what is probability that the card is either an ace or spade?

Let $X$ be the event of picking an ace and $Y$ be the event of picking a spade.

$P (X) = \frac{4}{52}$

$P (Y) = \frac{13}{52}$

The two events are not mutually exclusive, as there is one favorable outcome in which the card can be both an ace and spade.

$P (X and Y) = \frac{1}{52}$

$\begin{array}{l} P (X or Y) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} \\ = \frac{16}{52} \\ = \frac{4}{13} \end{array}$