Addition Law

The addition law of probability (sometimes referred to as the addition rule or sum rule), states that the probability that $A$ or $B$ will occur is the sum of the probabilities that $A$ will happen and that $B$ will happen, minus the probability that both $A$ and $B$ will happen. The addition rule is summarized by the formula:

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

Consider the following example. When drawing one card out of a deck of $52$ playing cards, what is the probability of getting heart or a face card (king, queen, or jack)? Let $H$ denote drawing a heart and $F$ denote drawing a face card. Since there are $13$ hearts and a total of $12$ face cards ($3$ of each suit: spades, hearts, diamonds and clubs), but only $3$ face cards of hearts, we obtain:

$\displaystyle P(H) = \frac{13}{52}$

$\displaystyle P(F) = \frac{12}{52}$

$\displaystyle P(F \cap H) = \frac{3}{52}$

Using the addition rule, we get:

$\displaystyle \begin{aligned} P(H\cup F)&=P(H)+P(F)-P(H\cap F)\\ &=\frac { 13 }{ 52 } +\frac { 12 }{ 52 } -\frac { 3 }{ 52 } \end{aligned}$

The reason for subtracting the last term is that otherwise we would be counting the middle section twice (since $H$ and $F$ overlap). 

Addition Rule for Disjoint Events

Suppose $A$ and $B$ are disjoint, their intersection is empty. Then the probability of their intersection is zero. In symbols: $P(A \cap B) = 0$. The addition law then simplifies to:

$P(A \cup B) = P(A) + P(B) \qquad \text{when} \qquad A \cap B = \emptyset$ 

The symbol $\emptyset$ represents the empty set, which indicates that in this case $A$ and $B$ do not have any elements in common (they do not overlap).

Example:

Suppose a card is drawn from a deck of 52 playing cards: what is the probability of getting a king or a queen? Let $A$ represent the event that a king is drawn and $B$ represent the event that a queen is drawn. These two events are disjoint, since there are no kings that are also queens. Thus:

$\displaystyle \begin{aligned} P(A \cup B) &= P(A) + P(B)\\&=\frac{4}{52}+\frac{4}{52}\\&=\frac{8}{52}\\&=\frac{2}{13} \end{aligned}$