Basic Probability Rules

The probability of any event #A# occurring, #\mathbb{P}(A)#, is always greater than or equal to #0# and less than
or equal to #1#:
\[0\leq \mathbb{P}(A) \leq 1\]

The sum of the probabilities of all possible outcomes must equal #1#. That is, if the sample space is #\Omega#, then:
\[\sum_{\text{all }x \text{ in }\Omega}\mathbb{P}(x)=1\]

Probability of the Complement

The probability that event #A# occurs plus the probability that its complement #A^c# occurs is always equal to #1#:

• #\mathbb{P}(A) + \mathbb{P}(A^c) = 1#
• #\mathbb{P}(A^c) = 1 - \mathbb{P}(A)#
• #\mathbb{P}(A) = 1 - \mathbb{P}(A^c)#

Conditional Probabilities

To calculate the conditional probability of #A# given #B#, make use of the following rules:

• If #A# and #B# have overlapping outcomes, then:
  \[\mathbb{P}(A\,|\,B)=\cfrac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}\]
• If #A# and #B# are mutually exclusive events, then:
  \[\mathbb{P}(A\,|\,B) = 0\]

Probability of the Intersection

To calculate the probability of the intersection of two events #A# and #B#, make use of the following rules:

• If #A# and #B# are dependent events, then:
  \[\mathbb{P}(A \cap B) = \mathbb{P}(A\,|\,B)\cdot \mathbb{P}(B) = \mathbb{P}(B\,|\,A)\cdot \mathbb{P}(A)\]
• If #A# and #B# are independent events, then:
  \[\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B)\]
• If #A# and #B# are mutually exclusive events, then:
  \[\mathbb{P}(A \cap B) = 0\]

Probability of the Union

To calculate the probability of the union of two events #A# and #B#, make use of the following rules:

• If #A# and #B# have overlapping outcomes, then:
  \[\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) − \mathbb{P}(A \cap B)\]
• If #A# and #B# are mutually exclusive events, then:
  \[\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B)\]

Probability of the Difference

To calculate the probability of the difference of two events #A# and #B#, make use of the following rules:

• If #A# and #B# have overlapping outcomes, then:
  \[\mathbb{P}(A \,\backslash\, B) = \mathbb{P}(A) − \mathbb{P}(A \cap B)\]
• If #A# and #B# are mutually exclusive events, then:
  \[\mathbb{P}(A \,\backslash\, B) = \mathbb{P}(A)\]

Law of Total Probability

Given a partition of the sample space #A_1, A_2, \ldots, A_k# and some other event #B#, it is possible to define event #B# as:

#\phantom{0}#
Using this definition, the probability of event #B# occurring can be calculated with the Law of Total Probability:

\[\mathbb{P}(B) =\mathbb{P}(B\,|\,A_1)\cdot \mathbb{P}(A_1) + \mathbb{P}(B\,|\,A_2)\cdot \mathbb{P}(A_2) + \ldots + \mathbb{P}(B\,|\,A_k)\cdot \mathbb{P}(A_k)\]

Bayes' Theorem

Bayes’ Theorem is stated mathematically as the following equation:

\[\mathbb{P}(A\,|\,B) = \frac{\mathbb{P}(B|A)\cdot \mathbb{P}(A)}{\mathbb{P}(B)}\]