Definition of Probability

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

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

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:

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