math

I. What is Probability?

Definition: Probability measures how likely something is to happen.

Formula:

P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

Examples:

• Flipping a coin → P(Heads) = 1/2

• Rolling a die → P(rolling a 3) = 1/6

II. Complement Rule

Definition: The complement is the probability that an event does not happen.

Formula:

P(A{\prime}) = 1 - P(A)

Examples:

• P(rolling a 1 on a die) = 1/6 → P(not rolling a 1) = 1 - 1/6 = 5/6

• P(it rains) = 0.3 → P(no rain) = 1 - 0.3 = 0.7

III. Addition Rule (For “OR” Statements)

General Rule (when events might overlap):

P(A \cup B) = P(A) + P(B) - P(A \cap B)

If events are mutually exclusive (can’t both happen):

P(A \cup B) = P(A) + P(B)

Keywords: “either,” “or,” “at least one”

Example:

• P(Popsicle) = 0.27, P(Chocolate) = 0.35, P(Both) = 0.15
  P(Popsicle \cup Chocolate) = 0.27 + 0.35 - 0.15 = 0.47

IV. Multiplication Rule (For “AND” Statements)

For Independent Events:

P(A \cap B) = P(A) \times P(B)

For Dependent Events (Conditional):

P(A \cap B) = P(A | B) \times P(B)

Keywords: “and,” “both,” “at the same time”

Example (independent):

• P(rolling a 3) = 1/6, P(flipping heads) = 1/2
  P(3 \text{ and } Heads) = 1/6 \times 1/2 = 1/12

V. Conditional Probability

Definition: The probability that event A happens given that B already happened.

Formula:

P(A | B) = \frac{P(A \cap B)}{P(B)}

Use when the problem gives a “given that” or “if B happens” clue.

Example:

• P(Biology) = 0.70, P(Algebra | Biology) = 0.40
  P(Algebra \cap Biology) = 0.40 \times 0.70 = 0.28

VI. Independent vs. Mutually Exclusive

Independent:

• Events don’t affect each other

• P(A | B) = P(A), and P(B | A) = P(B)

• Multiply to find “and”

Mutually Exclusive:

• Events cannot happen together

• P(A and B) = 0

• Use simple addition for “or”

How to Tell the Difference:

• If one event happening changes the chances of the other → Not independent

• If both can’t happen at the same time → Mutually exclusive

VII. Venn Diagrams & Set Notation

Symbols to Know:

• ∪ = Union = OR

• ∩ = Intersection = AND

• ′ = Complement = NOT

Formulas:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

P(A{\prime}) = 1 - P(A)

How to Use Venn Diagrams:

1. Start with the overlap (A ∩ B)

2. Fill in the rest of each circle

3. Use the outside area for the complement

VIII. Word Problem Templates

Example 1: Classes

Sarah takes Algebra (P = 0.30), Biology (P = 0.70), and both (given: P(Algebra | Biology) = 0.40)

• P(Both) = 0.40 × 0.70 = 0.28

• P(Algebra or Biology) = 0.30 + 0.70 – 0.28 = 0.72

• Not independent (0.30 ≠ 0.40), not mutually exclusive (P(Both) ≠ 0)

Example 2: Ridgemont High

150 students: 70 have dogs, 30 have cats, 25 have both

• P(Dog ∩ Cat) = 25/150 = 1/6

• P(Dog ∪ Cat) = (70 + 30 – 25)/150 = 75/150 = 1/2

• P(Neither) = 1 – 75/150 = 1/2

Example 3: Complements

• P(rain 3 days) = ?

• Complement = P(not raining 3 days) = 1 – P(rain 3 days)

IX. Key Phrases and What They Mean

X. Final Tips

• Draw diagrams when the problem mentions percentages or sets

• Label your work clearly

• Know your formulas but look for keywords: or, and, given, not

• Check if it’s asking for BOTH, EITHER, or NOT

• Bring your index card and take your time!