Addition Theorem
The addition theorem states that if two events A and B are mutually exclusive the probability of the occurrence of either \(A\) or \(B\) is the sum of the individual probability of \(A\) and \(B\). Symbolically, \[P(A\ \text{or}\ B)=P(A)+P(B)\] Proof:
If an event \(A\) can happen in \(a_1\) ways and \(B\) can happen in \(a_2\) ways, then the number of ways in which either event can happen is \(a_1+a_2\). If the total number of possibilities is \(n\), then by definition the probability of either the first or the second event happening is \[\frac{a_1+a_2}{n}=\frac{a_1}{n}+\frac{a_2}{n}\] But, \[\frac{a_1}{n}=P(A)\] and \[\frac{a_2}{n}=P(B)\] Hence,
\[P(A\ \text{or}\ B)=P(A) + P(B)\]
Examples
Example:
One card is drawn from a standard pack of 52. What is the probability that it is wither a king or a queen?
Solution:
There are 4 kings and 4 queens in a pack of 52 cards.
\(\therefore\) The probability that the card is drawn is a king = \(\frac{4}{52}\)
and the probability that the card drawn is a queen = \(\frac{4}{52}\)
Since, the events are mutually exclusive, the probability that the card drawn is either a king or a queen \[= \frac{4}{52}+\frac{4}{52}=\frac{8}{52}=\frac{2}{13}\]
When events are NOT mutually exclusive
\[P(A\ \text{or}\ B)=P(A) + P(B) - P(A\ \text{and}\ B)\]
Example
The probability of drawing a king or a heart shall be: \[P\ (\text{king\ } \text{or } \text{heart}) =P\ (\text{king})+P\ (\text{heart}) - P(\text{king\ and\ heart})\] \[= \frac{4}{52}+\frac{13}{52}-\frac{1}{52}=\frac{16}{52}=\frac{4}{13}\]
\(P(A\ \text{or}\ B)\) can be written as \(A \cup B\)