Chapter 4: Probability

Probability

The chance that an event will occur

Experiment

An activity that produces outcomes

Outcome

A possible result of an experiment

Sample Space

The set of all possible outcomes

Event

A set of outcomes from the sample space

Equally Likely Outcomes

All outcomes have the same probability of occurring

Theoretical Probability

Probability based on equally likely outcomes; favorable outcomes divided by total outcomes

Experimental Probability

Probability based on actual results from trials

Subjective Probability

Probability based on personal judgment or experience

Probability Formula

P(A) = favorable outcomes ÷ total outcomes

Law of Large Numbers

As the number of trials increases, experimental probability approaches theoretical probability

Complement

An event not occurring

Complement Formula

P(A') = 1

Conditional Probability

The probability of event A occurring given that event B has already occurred

Conditional Probability Notation

P(A|B)

Conditional Probability Formula

P(A|B) = P(A AND B) ÷ P(B)

AND Event

An outcome that is in both event A and event B

AND Symbol

∩

P(A AND B)

The probability that both A and B occur

Multiplication Rule

P(A AND B) = P(B) × P(A|B)

Independent Events

Two events where one event does not affect the probability of the other

Independent Events Formula

P(A AND B) = P(A) × P(B)

How to Check Independence

P(A|B) = P(A) or P(A AND B) = P(A)P(B)

Dependent Events

Two events where one event affects the probability of the other

With Replacement

The selected item is returned before the next selection; events are independent

Without Replacement

The selected item is not returned before the next selection; events are dependent

OR Event

An outcome that is in A or B or both

OR Symbol

∪

Mutually Exclusive Events

Two events that cannot occur at the same time

Mutually Exclusive Rule

P(A AND B) = 0

Addition Rule

P(A OR B) = P(A) + P(B)

Addition Rule for Mutually Exclusive Events

P(A OR B) = P(A) + P(B)

Why Subtract P(A AND B) in the Addition Rule

To avoid counting the overlap twice

P(A OR B)

The probability that A occurs, B occurs, or both occur

Counting Rules

Methods used to count the number of possible outcomes

Multiplication Rule of Counting

If one event occurs in m ways and another occurs in n ways, both can occur in m × n ways

Permutation

An arrangement where order matters

When to Use Permutations

When positions or order are important

Permutation Calculator Function

nPr

Combination

A selection where order does not matter

When to Use Combinations

When positions or order are not important

Combination Calculator Function

nCr

Example of a Permutation

Choosing a president, vice president, and secretary

Example of a Combination

Selecting 7 people from 20 for a survey

Independent vs Mutually Exclusive

Independent events can occur together; mutually exclusive events cannot

Keyword AND

Use multiplication rule

Keyword OR

Use addition rule

Keyword GIVEN

Use conditional probability

Keyword NOT

Use complement rule

Keyword Order Matters

Use permutation (nPr)

Keyword Order Does Not Matter

Use combination (nCr)

Keyword Independent

Check whether P(A AND B) = P(A)P(B)

Keyword Mutually Exclusive

Check whether P(A AND B) = 0