8.1-8.4: Probability Notes:

Understanding the Basics of Chance and Likelihood

Key Concepts

1. Probability Scale

• Probability is measured on a scale from 0 to 1:

  • 0 means an event is impossible

  • 1 means an event is certain

  • Values between 0 and 1 show how likely something is to happen

2. Single Event Probability

• Probability of an event happening = Number of successful outcomes ÷ Total number of possible outcomes

• Example: If a die is rolled, the chance of getting a 4 is 1 ÷ 6 = 0.166... ≈ 17%

3. Complementary Events

• The probability of something not happening = 1 – Probability of it happening

• Example: If the chance of rain is 0.3, then the chance of no rain is 1 – 0.3 = 0.7

How to Present Answers

• You can write probabilities as:

  • Fractions (e.g. ⅔)

  • Decimals (e.g. 0.67)

  • Percentages (e.g. 67%)

Examples:

• 0 → Getting a 7 on a standard die

• 0.5 → Tossing a coin and getting heads

• 1 → The sun rising tomorrow

Estimating Probability from Experiments and Predicting Outcomes

Key Concepts

1. Relative Frequency

• This is an estimate of probability based on experimental results

• Formula: Relative frequency = Number of times an outcome occurs ÷ Total number of trials

• Example: If a spinner lands on red 18 times out of 60 spins: Relative frequency of red = 18 ÷ 60 = 0.3

2. Expected Frequency

• This predicts how often an event will occur in a larger number of trials

• Formula: Expected frequency = Probability × Total number of trials

• Example: If the probability of rolling a 6 is 1/6, then in 120 rolls: Expected frequency of rolling a 6 = 1/6 × 120 = 20

3. Fair vs Biased vs Random

• Fair: All outcomes have equal chance e.g. tossing a standard coin

• Biased: Some outcomes are more likely than others e.g. a spinner with unequal-sized sections

• Random: Outcomes are unpredictable but follow probability patterns over time e.g. drawing names from a hat

Using Diagrams to Calculate Multi-Step Probabilities

Key Concepts

1. Combined Events

• These involve more than one outcome or step

• Example: Tossing two coins, or rolling a die twice

2. With Replacement

• After each event, the item is returned before the next draw

• Probabilities stay the same for each step

Tools for Solving Combined Probabilities

1. Sample Space Diagrams

• Show all possible outcomes

• Useful for simple events like rolling two dice

• Example:

• Outcomes for rolling two dice: (1,1), (1,2), ..., (6,6)

2. Venn Diagrams (limited to two sets)

• Show overlapping probabilities

• Use when events share outcomes

• Example:

  • Set A = students who like math

  • Set B = students who like science

  • A ∩ B = students who like both

3. Tree Diagrams

• Show branching outcomes step-by-step

• Probabilities are written beside each branch

• Final outcomes are at the ends of branches

• Multiply probabilities along branches to find combined probability

Probability Problems with Algebraic and Diagrammatic Tools

Algebraic Probability

• Use algebra to represent unknown probabilities

• Solve equations involving probabilities

• Example: 

  • If the probability of A is x and the probability of B is 2x, and total probability is 1: 

→ x + 2x = 1 → 3x = 1 → x = 1⁄3

Combined Events with Algebra

• Use algebraic expressions in tree diagrams or Venn diagrams

• Example:

• A bag has red and blue counters. Probability of red = x → Probability of blue = 1 – x. If you draw twice with replacement, then:

• First draw:

  • Red: x

  • Blue: 1 – x

• Second draw:

  • Red after red: x × x = x²

  • Blue after red: x × (1 – x) = x(1 – x)

• You can use this to calculate probabilities of sequences like “red then blue.”

Conditional Probability

• Probability of an event occurring given that another has already occurred

• Formula: P(A | B) = P(A ∩ B) ÷ P(B)

Explained in simple terms and steps:

1. Create a contingency table

2. Calculate the probability for each part of the table

3. Analyze the question

Example: