Definition

Theoretical probability is based on what we expect to happen in theory, without actually performing the experiment.

Formula

P(Event) = Number of Favorable Outcomes / Total Number of Possible Outcomes

📝 Example

What is the probability of rolling a 4 on a fair die?

• Favorable outcomes: 1 (only one face shows 4)

• Total possible outcomes: 6

P(4) = 1/6 ≈ 0.167 or 16.7%

Definition

Experimental probability is based on what actually happens when we conduct an experiment or collect data.

Formula

P(Event) = Number of Times Event Occurred / Total Number of Trials

📝 Example

A coin was flipped 50 times. Heads appeared 28 times. What is the experimental probability?

• Event occurred: 28 times

• Total trials: 50

P(Heads) = 28/50 = 0.56 or 56%

Note: Theoretical probability is 0.5, but experimental may differ!

Independent Events

Independent events: The occurrence of one event does NOT affect the probability of the other event.

P(A and B) = P(A) × P(B)

Examples: Flipping two coins, rolling two dice

Dependent Events

Dependent events: The occurrence of one event DOES affect the probability of the other event.

P(A and B) = P(A) × P(B|A)

Example: Drawing two cards without replacement

📝 Example

Independent:

Roll a die twice. P(6 first, then 5) = P(6) × P(5) = (1/6) × (1/6) = 1/36

Dependent:

Draw 2 cards from a deck without replacement.

P(Ace, then King) = (4/52) × (4/51) = 16/2652 ≈ 0.006

Definition

Conditional probability is the probability of event A occurring, given that event B has already occurred. Written as P(A|B), read as "probability of A given B."

Formula

P(A|B) = P(A ∩ B) / P(B)

where P(B) > 0

📝 Example

In a class, 60% study math, 40% study physics, and 25% study both. If a student studies math, what's the probability they also study physics?

• P(Math and Physics) = 0.25

• P(Math) = 0.60

P(Physics|Math) = 0.25 / 0.60 ≈ 0.417 or 41.7%

Definition

If there are m ways to do one thing and n ways to do another, then there are m × n ways to do both.

Formula

Total Outcomes = n₁ × n₂ × n₃ × ... × nₖ

where n₁, n₂, n₃, ... nₖ are the number of choices for each event

📝 Example

A restaurant offers 4 appetizers, 6 main courses, and 3 desserts. How many different three-course meals are possible?

Total meals = 4 × 6 × 3 = 72 different meals

Definition

A permutation is an arrangement of objects where order matters.

Formulas

Permutation of n objects taken r at a time:

nPr = n! / (n - r)!

Permutation of all n objects:

nPn = n!

Where n! = n × (n-1) × (n-2) × ... × 2 × 1

📝 Example

How many ways can 5 students be arranged in a row for a photo?

5P5 = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways

How many ways can we select and arrange 3 out of 5 students?

5P3 = 5!/(5-3)! = 5!/2! = 120/2 = 60 ways

Definition

A combination is a selection of objects where order does NOT matter.

Formula

Combination of n objects taken r at a time:

nCr = n! / [r! × (n - r)!]

Relationship: Permutations and Combinations

nPr = r! × nCr

Permutations = (Combinations) × (Ways to arrange r items)

📝 Example

Choose 3 students from a group of 5 for a committee (order doesn't matter).

5C3 = 5! / [3! × 2!]

= (5 × 4 × 3 × 2 × 1) / [(3 × 2 × 1) × (2 × 1)]

= 120 / (6 × 2) = 120 / 12

= 10 ways

What is a Two-Way Table?

A two-way frequency table displays data for two categorical variables, with one variable in rows and the other in columns.

Key Terms

• Joint Frequency: Values in the interior cells

• Marginal Frequency: Row and column totals

• Total Frequency: Grand total of all data

Finding Probabilities

P(Event) = Frequency of Event / Total Frequency

📝 Example Table

P(Male and Plays Sports) = 30/100 = 0.30

P(Plays Sports | Female) = 25/50 = 0.50

For Mutually Exclusive Events

Mutually exclusive events cannot occur at the same time.

P(A or B) = P(A) + P(B)

Example: Rolling a 3 or a 5 on a die (can't get both)

For Non-Mutually Exclusive Events

Non-mutually exclusive events can occur at the same time.

P(A or B) = P(A) + P(B) - P(A and B)

Subtract P(A and B) to avoid double-counting the overlap

📝 Example

A card is drawn from a deck. Find P(King or Heart).

• P(King) = 4/52

• P(Heart) = 13/52

• P(King and Heart) = 1/52 (King of Hearts)

P(King or Heart) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13

Definition

Geometric probability uses geometric measures (length, area, volume) to find probabilities.

Formulas

For Length:

P = Length of Favorable Region / Total Length

For Area:

P = Area of Favorable Region / Total Area

📝 Example

A dartboard is a circle with radius 10 cm. The bullseye is a circle with radius 2 cm. What's the probability of hitting the bullseye?

• Area of bullseye = π(2)² = 4π cm²

• Total area = π(10)² = 100π cm²

P(Bullseye) = 4π / 100π = 4/100 = 0.04 or 4%