Probability and Counting Principles

Fundamental Counting Principle

The Fundamental Counting Principle is used to determine the total number of possible outcomes in a sequence of events. If one event can occur in m ways and a second event can occur in n ways, then the two events together can occur in m × n ways.

• Key Point: Multiply the number of choices for each event.

• Example: If you have 3 shirts and 4 pants, the number of possible outfits is 3 × 4 = 12.

Permutations and Combinations

Permutations count arrangements where order matters; combinations count selections where order does not matter.

• Permutation Formula:

• Combination Formula:

• Example (Permutation): Arranging 3 books out of 5 on a shelf:

• Example (Combination): Choosing 3 students out of 5:

Probability Concepts

Basic Probability Rules

• Probability of an Event: is the likelihood that event E occurs.

• Complement Rule: , where E' is the complement of E.

• Multiplication Rule:

• Addition Rule:

• Conditional Probability:

Example: If P(A) = 0.3, P(B) = 0.4, and P(A and B) = 0.1, then P(A or B) = 0.3 + 0.4 - 0.1 = 0.6.

Probability from Tables and Frequency Distributions

Probabilities can be calculated using data from tables or frequency distributions by dividing the number of favorable outcomes by the total number of outcomes.

• Example: If a table shows 20 students, and 5 received an A, then .

Conditional Probability

Conditional probability measures the probability of an event occurring given that another event has already occurred.

• Formula:

• Example: If 10 students are left-handed and 4 are female, and 2 are both, then .

Probability Involving "Or" (Addition Rule)

To find the probability that at least one of several events occurs, use the addition rule.

• Formula:

• Example: Probability of drawing a heart or a king from a deck of cards: , , , so .

Probability Involving Multiple Trials or Selections

When an experiment involves several trials or selections, use the multiplication rule or binomial probability as appropriate.

• Example: Probability of flipping two heads in two coin tosses: , so .

Discrete and Continuous Random Variables

Discrete vs. Continuous Random Variables

A discrete random variable takes on countable values (e.g., number of students), while a continuous random variable takes on any value in an interval (e.g., height).

• Discrete Example: Number of cars in a parking lot (0, 1, 2, ...)

• Continuous Example: Time to run a mile (can be any positive real number)

Discrete Probability Distributions

Mean, Variance, and Standard Deviation

For a discrete probability distribution, the mean (expected value), variance, and standard deviation are calculated as follows:

• Mean:

• Variance:

• Standard Deviation:

• Example: For x = 0, 1, 2 with P(x) = 0.2, 0.5, 0.3:

Binomial Distribution

Binomial Probability Formula

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Formula: , where

• Mean:

• Variance:

• Standard Deviation:

• Example: Probability of 2 heads in 3 coin tosses ():

Normal Distribution

Standard Normal Distribution and Z-Scores

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Probabilities are found using z-scores and standard normal tables.

• Z-Score Formula:

• Finding Areas: Use the z-score to look up probabilities in the standard normal table.

• Example: If , , , then .

Summary Table: Key Probability Formulas

Additional info: These notes synthesize the exam review topics and formulas, expanding on each with definitions, examples, and context for introductory statistics students.