Probability, Counting, and Discrete Distributions: Exam 2 Review Notes

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: $nP_r = \frac{n!}{(n-r)!}$

• Combination Formula: $nC_r = \frac{n!}{r!(n-r)!}$

• Example (Permutation): Arranging 3 books out of 5 on a shelf: $5P_3 = \frac{5!}{2!} = 60$

• Example (Combination): Choosing 3 students out of 5: $5C_3 = \frac{5!}{3!2!} = 10$

Probability Concepts

Basic Probability Rules

• Probability of an Event: $P(E)$ is the likelihood that event E occurs.

• Complement Rule: $P(E') = 1 - P(E)$, where E' is the complement of E.

• Multiplication Rule: $P(A \text{ and } B) = P(A)P(B|A)$

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

• Conditional Probability: $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$

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 $P(\text{A}) = \frac{5}{20} = 0.25$.

Conditional Probability

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

• Formula: $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$

• Example: If 10 students are left-handed and 4 are female, and 2 are both, then $P(\text{Female}|\text{Left-handed}) = \frac{2}{10} = 0.2$.

Probability Involving "Or" (Addition Rule)

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

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

• Example: Probability of drawing a heart or a king from a deck of cards: $P(\text{Heart}) = \frac{13}{52}$, $P(\text{King}) = \frac{4}{52}$, $P(\text{Heart and King}) = \frac{1}{52}$, so $P(\text{Heart or King}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52}$.

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: $P(\text{Head}) = 0.5$, so $P(\text{Head and Head}) = 0.5 \times 0.5 = 0.25$.

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: $\mu = \sum x \cdot P(x)$

• Variance: $\sigma^2 = \sum (x - \mu)^2 \cdot P(x)$

• Standard Deviation: $\sigma = \sqrt{\sigma^2}$

• Example: For x = 0, 1, 2 with P(x) = 0.2, 0.5, 0.3: $\mu = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 0 + 0.5 + 0.6 = 1.1$

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: $P(x) = nC_x p^x q^{n-x}$, where $q = 1 - p$

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

• Standard Deviation: $\sigma = \sqrt{npq}$

• Example: Probability of 2 heads in 3 coin tosses ($p = 0.5$): $P(2) = 3C_2 (0.5)^2 (0.5)^{1} = 3 \times 0.25 \times 0.5 = 0.375$

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: $z = \frac{x - \mu}{\sigma}$

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

• Example: If $x = 85$, $\mu = 80$, $\sigma = 5$, then $z = \frac{85 - 80}{5} = 1.0$.

Summary Table: Key Probability Formulas

Formula	Description
$P(E') = 1 - P(E)$	Probability of the complement of an event
$P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$	Conditional probability
$P(A \text{ and } B) = P(A)P(B|A)$	Multiplication Rule for probability
$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$	Addition Rule for probability
$\mu = \sum x \cdot P(x)$	Mean of a discrete probability distribution
$\sigma^2 = \sum (x - \mu)^2 \cdot P(x)$	Variance of a discrete probability distribution
$P(x) = nC_x p^x q^{n-x}$	Binomial probability formula
$\mu = np$	Mean of a binomial distribution
$\sigma^2 = npq$	Variance of a binomial distribution

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