Addition and Multiplication Rules in Probability

Chapter 4: Probability

Addition Rule and Multiplication Rule

This section introduces the fundamental rules for calculating probabilities involving two or more events. Understanding these rules is essential for analyzing compound events and interpreting the likelihood of various outcomes in statistics.

Learning Objectives

• To find and interpret probabilities for two or more events using the addition and multiplication rules.

• To identify disjoint (mutually exclusive) and independent events.

Key Terms

• Compound Event: An event that combines two or more simple events.

• Disjoint (Mutually Exclusive) Events: Events that cannot occur at the same time. If one occurs, the other cannot.

• Independent Events: Events where the occurrence of one does not affect the probability of the other.

Compound Events

A compound event is any event that combines two or more simple events. Understanding the relationships between these events is crucial for applying probability rules correctly.

• Disjoint Events: Two events are disjoint if they cannot both occur in the same trial. For example, selecting a person who is male and a person who is female in a single selection are disjoint events.

• Independent Events: Two events are independent if the occurrence of one does not influence the probability of the other. For example, flipping a coin and rolling a die are independent events.

Addition Rule

The addition rule is used to find the probability that either event A or event B occurs (or both). The word "or" in probability typically indicates addition.

• Intuitive Addition Rule: To find $P(A \text{ or } B)$, add the number of ways event A can occur and the number of ways event B can occur, but ensure that outcomes common to both are only counted once.

• Formal Addition Rule:

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Here, $P(A \text{ and } B)$ is the probability that both events occur simultaneously in a single trial.

Examples

• Disjoint Events Example: Event A: Randomly selecting someone for a clinical trial who is a male. Event B: Randomly selecting someone for a clinical trial who is a female. The selected person cannot be both.

• Not Disjoint Events Example: Event A: Randomly selecting someone taking a statistics course. Event B: Randomly selecting someone who is a female. The selected person can be both.

Summary of the Addition Rule

• Associate the word "or" with addition.

• When calculating $P(A \text{ or } B)$, add the number of ways A can occur and the number of ways B can occur, but avoid double-counting outcomes that are common to both events.

Complementary Events and the Addition Rule

The complement of an event A, denoted as $A'$, is the event that A does not occur. The sum of the probabilities of an event and its complement is always 1.

• $P(A) + P(A') = 1$

• $P(A') = 1 - P(A)$

Example: If the probability that a randomly selected person has sleepwalked is 0.292, then the probability that a person has not sleepwalked is $1 - 0.292 = 0.708$.

Multiplication Rule

The multiplication rule is used to find the probability that two events A and B both occur (i.e., $P(A \text{ and } B)$). The word "and" in probability typically indicates multiplication.

• Intuitive Multiplication Rule: To find the probability that event A occurs in one trial and event B occurs in another, multiply the probability of A by the probability of B, adjusting for whether the events are independent or dependent.

• Formal Multiplication Rule:

$P(A \text{ and } B) = P(A) \times P(B|A)$

Here, $P(B|A)$ is the probability that B occurs given that A has already occurred.

Sampling and Independence

• Sampling with replacement: Each selection is independent.

• Sampling without replacement: Selections are dependent, as the outcome of one affects the next.

Example: Drug Screening

Suppose 45 out of 50 subjects have positive test results, and 5 have negative results. If two subjects are selected:

• With replacement: Probability that first is positive and second is negative:

$P(\text{positive first}) = \frac{45}{50}$ $P(\text{negative second}) = \frac{5}{50}$ $P(\text{positive first and negative second}) = \frac{45}{50} \times \frac{5}{50} = 0.0900$

• Without replacement: After selecting a positive, only 49 remain (5 negative):

$P(\text{positive first}) = \frac{45}{50}$ $P(\text{negative second}) = \frac{5}{49}$ $P(\text{positive first and negative second}) = \frac{45}{50} \times \frac{5}{49} = 0.0918$

Guideline for Cumbersome Calculations

When sampling without replacement from a large population, if the sample size is no more than 5% of the population, treat selections as independent for simplicity.

Example: If 3 adults are selected without replacement from 247,436,830 adults (10% use drugs), the probability all 3 use drugs is:

$P(\text{all 3 use drugs}) = (0.10)^3 = 0.00100$

Additional info: This uses the 5% guideline, as 3 is much less than 5% of the population.

Redundancy: Application of the Multiplication Rule

Redundancy increases system reliability by duplicating critical components. In probability, redundancy reduces the risk of total system failure.

• Example: Airbus 310 Hydraulic Systems

If the probability of a single hydraulic system failing is 0.002, then the probability it does not fail is:

$P(\text{no failure}) = 1 - 0.002 = 0.998$

With three independent systems, the probability all fail is:

$P(\text{all fail}) = (0.002)^3 = 0.000000008$

Thus, the probability that at least one system works (i.e., flight control is maintained) is:

$P(\text{at least one works}) = 1 - 0.000000008 = 0.999999992$

Using redundancy, the risk of failure is dramatically reduced, increasing safety.

Summary Table: Addition and Multiplication Rules

Rule	Keyword	Formula	When to Use
Addition Rule	or	$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$	Finding probability of either event A or B (or both) occurring
Multiplication Rule	and	$P(A \text{ and } B) = P(A) \times P(B|A)$	Finding probability of both event A and B occurring

Key Takeaways

• The addition rule is used for "or" probabilities; ensure not to double-count overlapping outcomes.

• The multiplication rule is used for "and" probabilities; account for dependence or independence between events.

• Redundancy in systems can be analyzed using the multiplication rule to assess reliability.