Addition Rule and Mutually Exclusive Events in Probability

Section 3.3: The Addition Rule in Probability

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time. If one event happens, the other cannot. Recognizing whether events are mutually exclusive is essential for applying the correct probability rules.

• Definition: Two events, A and B, are mutually exclusive if they have no outcomes in common.

• Example 1: Rolling a 3 on a die (Event A) and rolling a 4 on a die (Event B) are mutually exclusive, since a single roll cannot result in both outcomes.

• Example 2: Selecting a male student (Event A) and selecting a nursing major (Event B) are not mutually exclusive, as a student could be both male and a nursing major.

• Example 3: Selecting a blood donor with type O blood (Event A) and selecting a female blood donor (Event B) are not mutually exclusive, since a donor could be a female with type O blood.

The Addition Rule

The Addition Rule is used to find the probability that at least one of two events occurs. The formula depends on whether the events are mutually exclusive.

• General Addition Rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• For Mutually Exclusive Events: $P(A \text{ or } B) = P(A) + P(B)$ Since $P(A \text{ and } B) = 0$ for mutually exclusive events.

• Extension: The rule can be extended to any number of mutually exclusive events by summing their individual probabilities.

Examples: Using the Addition Rule

• Example 1: Drawing a 4 or an ace from a standard deck of 52 cards.

  • Events are mutually exclusive (a card cannot be both a 4 and an ace).

  • Calculation: $P(4 \text{ or } A) = P(4) + P(A) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = 0.15$

• Example 2: Rolling a number less than 3 or an odd number on a die.

  • Events are not mutually exclusive (the outcome 1 is both less than 3 and odd).

  • Calculation: $P(\text{less than 3 or odd}) = P(\text{less than 3}) + P(\text{odd}) - P(\text{less than 3 and odd})$

A Summary of Probability

This section summarizes key probability concepts and rules, providing a foundation for solving probability problems.

Type	Description	Formula
Classical Probability	All outcomes are equally likely.	$P(E) = \frac{\text{Number of outcomes in event E}}{\text{Number of outcomes in sample space}}$
Empirical Probability	Estimated from experimentation or observation.	$P(E) = \frac{\text{Frequency of event E}}{\text{Total frequency}}$
Range Rule	Probability is always between 0 and 1, inclusive.	$0 \leq P(E) \leq 1$
Complementary Events	The complement of event E is all outcomes not in E.	$P(E') = 1 - P(E)$
Multiplication Rule	Probability of two events occurring in sequence.	$P(A \text{ and } B) = P(A) \cdot P(B|A)$ (dependent) $P(A \text{ and } B) = P(A) \cdot P(B)$ (independent)
Addition Rule	Probability of at least one of two events occurring.	$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Combining Rules to Find Probabilities

Sometimes, probability problems require combining the addition and complement rules. For example, to find the probability that a randomly selected NFL draft pick is not a running back or wide receiver, first find the probability of selecting a running back or wide receiver, then subtract from 1.

• Define Events:

  • A: Draft pick is a running back

  • B: Draft pick is a wide receiver

• Calculation:

  • Number of running backs: 16

  • Number of wide receivers: 37

  • Total picks: 255

  • $P(A \text{ or } B) = \frac{16}{255} + \frac{37}{255} = \frac{53}{255}$

  • Probability of not a running back or wide receiver: $P(\text{not } (A \text{ or } B)) = 1 - P(A \text{ or } B) = 1 - \frac{53}{255} = 0.79$