Addition Theorem of Probabilities
The sum $A + B$ of two events $A$ and $B$ is the event consisting of the occurrence of event $A$ or event $B$.
Theorem. The probability of occurrence of one of two mutually exclusive events equals the sum of their probabilities:
$P(A + B) = P(A) + P(B)$
Proof: $n$ - total number of outcomes, $m_a$ - outcomes favorable to $A$, $m_b$ - outcomes favorable to $B$
$P(A + B) = \frac{m_a + m_b}{n} = \frac{m_a}{n} + \frac{m_b}{n} = P(A) + P(B)$
Corollaries
- The sum of probabilities of all events $A_i \in \Omega$ forming a complete group of events equals one. $P(A_1) + … + P(A_n) = 1$
- The sum of the probability of event $A$ and its complementary event $\bar{A}$ equals one, since $A$ and $\bar{A}$ form a complete group of events. $P(A) + P(\bar{A}) = 1$
Addition Theorem for Compatible Events
Two events are called compatible if the occurrence of one does not exclude the occurrence of the other in the same trial.
Theorem. The probability of occurrence of at least one of two compatible events equals the sum of their probabilities minus the probability of their joint occurrence:
$P(A + B) = P(A) + P(B) - P(AB)$
Proof:
- $A + B$ occurs if $A\bar{B}$, $\bar{A}B$, or $AB$ occurs. Since these events are mutually exclusive, by the addition theorem we have $P(A + B) = P(A\bar{B}) + P(\bar{A}B) + P(AB)$ (*)
- $A$ occurs if either $AB$ or $A\bar{B}$ occurs. By the addition theorem, $P(A) = P(A\bar{B}) + P(AB)$ or $P(A\bar{B}) = P(A) - P(AB)$ (**)
- Similarly, $B$ occurs if either $AB$ or $\bar{A}B$ occurs. That is, $P(B) = P(\bar{A}B) + P(AB)$ or $P(\bar{A}B) = P(B) - P(AB)$ (***)
- Substituting (**) and (***) into (*), we get $P(A + B) = P(A) + P(B) - P(AB)$
Multiplication Theorem of Probabilities
The product of events $A$ and $B$ is the event $A \cdot B$ consisting of the joint occurrence of these events.
Example:
- $A$ - the part is functional
- $B$ - the part is painted
- $A \cdot B$ - the part is functional and painted
Theorem. Consider two events $A$ and $B$. We know $P(A)$ and $P(B\mid A)$. How do we find the probability that both $A$ and $B$ occur? $P(A \cdot B) = P(A) \cdot P(B \mid A)$ - by the definition of conditional probability. For independent events the multiplication theorem becomes
$P(A \cdot B) = P(A) \cdot P(B)$
See also
- Sum and Product Rules (Combinatorics)
Sources
- Gmurman V.E., Probability Theory and Mathematical Statistics – 9th edition. Moscow: Vysshaya Shkola, 2003.