### Formulas, Statistical Tables and R Commands: VVA Formula sheet

### VVA Formula sheet II (Probability)

### Definition and notation

Definition of Probability

To calculate the probability of event #A# occurring, make use of the following formula:

\[\mathbb{P}(A) = \cfrac{\text{Total number of outcomes classified as }A}{\text{Total number of outcomes in the sample space } \Omega}\]

Other notations:

- #\mathbb{P}(A^c)=# probability that the complement of #A# occurs
- #\mathbb{P}(A \cap B)=# probability that event #A#
*AND*event #B# occur together, the joint probability of #A# and #B# - #\mathbb{P}(A \cup B)=# probability that event #A#
*OR*event #B# occurs - #\mathbb{P}(A\,|\,B)=# probability of event #A# occurring
*GIVEN*that event #B# occurs, the probability of #A# conditional on #B# - #\mathbb{P}(A \,\backslash\, B)=# probability that event #A# occurs, but event #B# does
*NOT*

### Probability rules

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:

\[\sum_{\text{all }x \text{ in }\Omega}\mathbb{P}(x)=1\]

#\phantom{0}#

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)#

#\phantom{0}#

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:

\[\mathbb{P}(A\,|\,B) = 0\]

#\phantom{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:

\[\mathbb{P}(A \cap B) = 0\]

#\phantom{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:

\[\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B)\]

#\phantom{0}#

#\phantom{0}#

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:

\[\mathbb{P}(A \,\backslash\, B) = \mathbb{P}(A)\]

#\phantom{0}#

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:

\[B = (A_1 \cap B) \cup (A_2 \cap B) \cup \ldots \cup (A_k \cap B)\]

#\phantom{0}#

Using this definition, the probability of event #B# occurring can be calculated with the *Law of Total Probability*:

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

#\phantom{0}#

Bayes' Theorem

*Bayes’ Theorem* is stated mathematically as the following equation:

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