Computing Probabilities: Complement and Addition Rules

The complement of an event $E$ is the event that consists of all the outcomes not in $E$ (written as "NOT E").
Complement Rule Formula: For any event $E$ , $P(\text{NOT E}) = 1 - P(E)$ .
Example 2: In a prestatistics class, the probability of selecting a student with blue eyes ( $B$ ) is $0.3$ . The probability of selecting a student who does not have blue eyes is calculated as: $P(\text{NOT B}) = 1 - 0.3 = 0.7$

Disjoint Events: Also known as mutually exclusive events, these have no outcomes in common.
Addition Rule for Disjoint Events: If $E$ and $F$ are disjoint, then $P(E \text{ OR } F) = P(E) + P(F)$ .
Example 4: A six-sided die is rolled once.
- Event $E$ (even-number outcomes: $2, 4, 6$ ): $3$ of the $6$ outcomes are even, so $P(E) = \frac{3}{6}$ .
- Event $F$ (outcome $5$ ): This is $1$ of $6$ outcomes, so $P(F) = \frac{1}{6}$ .
- Because the number $5$ is not even, events $E$ and $F$ are disjoint.
- The probability of one OR the other occurring is: $P(E \text{ OR } F) = \frac{3}{6} + \frac{1}{6}$ .

General Addition Rule Formula: For any events $E$ and $F$ , $P(E \text{ OR } F) = P(E) + P(F) - P(E \text{ AND } F)$ .
This rule states that the probability of one event OR the other occurring is the sum of their probabilities minus the probability of both events occurring simultaneously.
Example 7: Rolling a six-sided die once to find the probability of rolling a number that is at least $3$ ( $T$ ) OR is even ( $E$ ).
- Outcomes for $T$ are: $3, 4, 5, 6$ .
- Outcomes for $E$ are: $2, 4, 6$ .
- Because outcomes $4$ and $6$ are in both $T$ and $E$ , the events are not disjoint.
- The general addition rule must be applied: $P(T \text{ OR } E) = P(T) + P(E) - P(T \text{ AND } E)$ .