Chapter 3: Probability

Sample Spaces and Events

In probability theory, a sample space is the set of all possible outcomes of a random experiment. An event is any subset of the sample space.

• Sample Space (S): The collection of all possible outcomes.

• Event (E): Any subset of the sample space, including the empty set and the sample space itself.

• Example: For tossing a coin, S = {Heads, Tails}. The event "getting Heads" is E = {Heads}.

Fundamental Counting Principle (Section 3.1)

The Fundamental Counting Principle states that 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.

• Formula: If there are k events, the total number of outcomes is

• Example: If you have 3 shirts and 2 pants, you have 3 × 2 = 6 possible outfits.

Theoretical and Empirical Probability (Section 3.1)

• Theoretical Probability: Based on reasoning or known possible outcomes.

  • Formula:

• Empirical Probability: Based on observations or experiments.

  • Formula:

• Example: If a die is rolled 60 times and lands on 4 fifteen times, the empirical probability of rolling a 4 is 15/60 = 0.25.

Conditional Probability and the Multiplication Rule (Section 3.2)

• Conditional Probability: The probability that event B occurs given that event A has occurred.

  • Formula:

• Multiplication Rule: The probability that both events A and B occur.

  • Formula:

• Example: If P(A) = 0.5 and P(B|A) = 0.3, then P(A and B) = 0.5 × 0.3 = 0.15.

Addition Rule (Section 3.3)

• Addition Rule for Any Two Events:

  • Formula:

• If A and B are mutually exclusive: , so

• Example: Probability of drawing a heart or a king from a deck of cards: P(heart) = 13/52, P(king) = 4/52, P(heart and king) = 1/52. So, .

Chapter 4: Discrete Probability Distributions

Constructing a Probability Distribution (Section 4.1)

A discrete probability distribution lists each possible value of a discrete random variable together with its probability.

• Each probability must be between 0 and 1.

• The sum of all probabilities must be 1.

• Example Table:

Mean, Variance, and Standard Deviation (Section 4.1)

• Mean (Expected Value):

• Variance:

• Standard Deviation:

• Example: For the table above,

Expected Value (Section 4.1)

• The expected value of a discrete random variable is the long-run average value of repetitions of the experiment it represents.

• It is calculated as the mean:

• Example: In a game where you win .

Binomial Distribution (Section 4.2)

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

• Conditions:

  • Fixed number of trials (n)

  • Each trial has two possible outcomes (success/failure)

  • Probability of success (p) is constant

  • Trials are independent

• Probability Formula:

• Mean:

• Variance:

• Standard Deviation:

• Example: Flipping a coin 5 times (n=5, p=0.5), probability of 3 heads:

Additional info: Poisson and geometric distributions are not included for this exam.

Chapter 5: Normal Probability Distributions

Normal Distributions and the Standard Normal Distribution (Section 5.1)

A normal distribution is a continuous, symmetric, bell-shaped distribution described by its mean (μ) and standard deviation (σ). The standard normal distribution is a normal distribution with μ = 0 and σ = 1.

• Equation for the normal curve:

• Many real-world phenomena are approximately normally distributed.

z-Scores and Areas Under the Standard Normal Curve (Section 5.1)

• z-Score: The number of standard deviations a value x is from the mean.

  • Formula:

• z-scores are used to find areas (probabilities) under the standard normal curve using tables or technology.

• Example: If μ = 100, σ = 15, x = 130, then

Finding Probabilities for Normal Distributions (Section 5.2)

• Convert x-values to z-scores, then use the standard normal table to find probabilities.

• Example: Probability that x < 130 when μ = 100, σ = 15: Find z = 2, look up P(z < 2) ≈ 0.9772.

Finding z-Scores Using Areas and Percentiles (Section 5.3)

• Given an area (probability), find the corresponding z-score using the standard normal table.

• Example: The 90th percentile corresponds to z ≈ 1.28 (since P(z < 1.28) ≈ 0.90).

Finding an x-Value Corresponding to a z-Score (Section 5.3)

• Given a z-score, solve for x using

• Example: If μ = 100, σ = 15, z = 1.28, then

Finding a Specific Data Value (Section 5.3)

• To find a data value corresponding to a given percentile, find the z-score for that percentile, then use

• Example: Find the value at the 95th percentile for μ = 80, σ = 10. z ≈ 1.645, so

Additional info: For the exam, you may bring one 8.5”x11” sheet of paper (front and back) for formulas/examples, and a calculator or phone.