Probability and Counting Rules

Sample Space

The set of all possible outcomes of an experiment.

Probability Rules

P(E) = Number of favorable outcomes ÷ Total number of outcomes.

Empirical Method

P(E) = Frequency of Event ÷ Total Number of Trials.

Mutually Exclusive Events

Events that cannot occur at the same time.

Addition Rule (Mutually Exclusive)

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

Addition Rule (Not Mutually Exclusive)

P(A or B) = P(A) + P(B) − P(A and B).

Complement of an Event

The probability that an event does not occur.

Complement Formula

P(A') = 1 − P(A).

Conditional Probability

The probability that event B occurs given that event A has already occurred.

Conditional Probability Formula

P(B|A) = P(A and B) ÷ P(A).

Independent Events

Events where the occurrence of one does not affect the probability of the other.

Multiplication Rule (Independent Events)

P(A and B) = P(A) × P(B).

Multiplication Rule (Dependent Events)

P(A and B) = P(A) × P(B|A).

P(At Least One)

P(at least one) = 1 − P(none).

Fundamental Principle of Counting

If a task consists of several stages, multiply the number of choices at each stage.

Permutation

An arrangement of objects where order matters.

Permutation Formula

nPr = n! ÷ (n − r)!.

Combination

A selection of objects where order does not matter.

Combination Formula

nCr = n! ÷ [r!(n − r)!].

OR Rule

Use the Addition Rule when calculating probabilities involving "or."

AND Rule

Use the Multiplication Rule when calculating probabilities involving "and."

Permutation Example

Choosing president, vice president, and secretary from a group.

Combination Example

Choosing 3 students from a class.

Sample Space Example

Rolling a die: {1, 2, 3, 4, 5, 6}.

Independent Event Example

Tossing a coin and rolling a die.