The Main Idea

1. Experiment: An activity with an observable result.
2. Outcome: A possible result of an experiment.
3. Sample Space: The set of all possible outcomes of an experiment.
4. Event: Any subset of a sample space.
5. Probability: The likelihood of an event occurring.
6. Probability Model: A mathematical description of an experiment listing all possible outcomes and their associated probabilities.

Probability Fundamentals

• Probability is always between [latex]0[/latex] and [latex]1[/latex] (or [latex]0\%[/latex] and [latex]100\%[/latex]).
  • [latex]0 \leq p \leq 1[/latex]
• [latex]0[/latex] ([latex]0\%[/latex]) indicates an impossible event.
• [latex]1[/latex] ([latex]100\%[/latex]) indicates a certain event.
• The sum of all probabilities in a probability model must equal [latex]1[/latex] ([latex]100\%[/latex]).

Calculating Probability

For equally likely outcomes:

[latex]P(\text{outcome}) = \frac{\text{Number of ways the outcome can occur}}{\text{Total number of possible outcomes}}[/latex]

Important: This formula only applies when all outcomes are equally likely to occur.

Constructing a Probability Model

1. Identify all possible outcomes (sample space).
2. Determine the total number of possible outcomes.
3. Calculate the probability of each outcome.
4. List each outcome with its associated probability.

1. Sample Space (S): The set of all possible outcomes in an experiment.
2. Event (E): Any subset of the sample space.
3. Equally Likely Outcomes: When all outcomes in the sample space have an equal chance of occurring.

Fundamental Formula

For an event [latex]E[/latex] in a sample space [latex]S[/latex] with equally likely outcomes:

[latex]P(E) = \frac{\text{number of elements in E}}{\text{number of elements in S}} = \frac{n(E)}{n(S)}[/latex]

Important: [latex]0 \leq P(E) \leq 1[/latex] always holds true, as [latex]E[/latex] is a subset of [latex]S[/latex].

Steps to Compute Probability

1. Identify the sample space [latex]S[/latex].
2. Count the total number of outcomes in [latex]S[/latex].
3. Identify the outcomes that make up the event [latex]E[/latex].
4. Count the number of outcomes in [latex]E[/latex].
5. Divide the number of outcomes in [latex]E[/latex] by the total number of outcomes in [latex]S[/latex].

The Main Idea

1. Union of Events: The event that occurs if either or both events occur.
2. Intersection of Events: The event that occurs when both events occur simultaneously.

Fundamental Formula

For two events [latex]E[/latex] and [latex]F[/latex]:

[latex]P(E \cup F) = P(E) + P(F) - P(E \cap F)[/latex]

Where:

• [latex]P(E \cup F)[/latex] is the probability of [latex]E[/latex] or [latex]F[/latex] occurring
• [latex]P(E \cap F)[/latex] is the probability of both [latex]E[/latex] and [latex]F[/latex] occurring

Important Notation

• Union: [latex]\cup[/latex] (represents “or”)
• Intersection: [latex]\cap[/latex] (represents “and”)

Steps to Compute Probability of Union

1. Calculate [latex]P(E)[/latex] and [latex]P(F)[/latex] individually
2. Determine [latex]P(E \cap F)[/latex]
3. Apply the formula: Add [latex]P(E)[/latex] and [latex]P(F)[/latex], then subtract [latex]P(E \cap F)[/latex]

The Main Idea

Mutually Exclusive Events: Events that cannot occur at the same time, i.e., they have no outcomes in common.

Simplified Formula

For mutually exclusive events [latex]E[/latex] and [latex]F[/latex]:

[latex]P(E \cup F) = P(E) + P(F)[/latex]

Note: This is a simplification of the general union formula because [latex]P(E \cap F) = 0[/latex] for mutually exclusive events.

Steps to Compute Probability of Mutually Exclusive Events

1. Verify that the events are mutually exclusive
2. Calculate [latex]P(E)[/latex] for the first event
3. Calculate [latex]P(F)[/latex] for the second event
4. Add the individual probabilities

The Main Idea

1. Complement of an Event: The set of all outcomes in the sample space that are not in the event.
2. Notation: The complement of event E is denoted as [latex]E'[/latex].

The Complement Rule

For any event [latex]E[/latex]:

[latex]P(E') = 1 - P(E)[/latex]

Where:

• [latex]P(E')[/latex] is the probability that event [latex]E[/latex] will not occur
• [latex]P(E)[/latex] is the probability that event [latex]E[/latex] will occur

Why It Works

The sum of the probabilities of all possible outcomes in a sample space must equal [latex]1[/latex]. Therefore, the probability of an event not occurring is the remainder after subtracting the probability of it occurring from [latex]1[/latex].

Steps to Find the Probability of an Event Not Happening

1. Identify the event [latex]E[/latex]
2. Calculate or determine [latex]P(E)[/latex]
3. Subtract [latex]P(E)[/latex] from [latex]1[/latex]

The Main Idea

1. Probability problems often involve counting principles, permutations, and combinations.
2. Break down complex problems into smaller counting problems.
3. Use combinations [latex](C(n,r))[/latex] to count ways of selecting items when order doesn’t matter.
4. Apply the Multiplication Principle when dealing with multiple independent selections.
5. Use the Complement Rule for “at least” problems.

General Approach

1. Identify the event you’re looking for.
2. Count the number of ways this event can occur using appropriate counting techniques.
3. Count the total number of possible outcomes (sample space).
4. Divide the number of favorable outcomes by the total number of possible outcomes.