Probability: Fundamental Concepts

Definitions and Notation

• Probability (P): A measure of how likely an event is to occur, expressed as a number between 0 and 1.

• Event (A, B, C): Any collection of results or outcomes of a procedure.

• Simple Event: An outcome that cannot be broken down into simpler components.

• Sample Space: The set of all possible simple events for a procedure.

• P(A): Probability of event A occurring.

Approaches to Probability

• Relative Frequency Approximation: Probability is approximated by repeating a procedure and observing the frequency of event A.

  • Formula:

• Classical Approach (Equally Likely Outcomes):

  • If a procedure has n equally likely simple events and event A can occur in s ways:

• Subjective Probability: Estimated using knowledge of relevant circumstances.

• Simulation: A process that mimics a real procedure to estimate probabilities when other methods are not feasible.

Rounding Probabilities

• Express probabilities as exact fractions or decimals, or round to three significant digits.

• Probabilities are typically not expressed as percentages in professional contexts.

Law of Large Numbers

• As the number of trials increases, the relative frequency probability approaches the actual probability.

• This law applies to large numbers of trials, not individual outcomes.

Cautions in Probability

• Do not assume outcomes are equally likely without justification.

• Avoid dividing numbers without understanding what they represent.

• Gamblers' fallacy: Past outcomes do not affect future probabilities in independent events.

Complementary Events

• The complement of event A (denoted as \( \overline{A} \)) consists of all outcomes where A does not occur.

• Formulas:

Significantly High or Low Results

• Significantly high: x successes among n trials is significantly high if .

• Significantly low: x successes among n trials is significantly low if .

• The threshold 0.05 is common but not absolute; other values (e.g., 0.01) may be used.

Probability Review

• Probability values range from 0 (impossible) to 1 (certain).

• Notation: for event A, for the complement.

Odds

• Odds against A: (expressed as a:b).

• Odds in favor of A: (reciprocal of odds against).

• Payoff odds: Ratio of net profit to amount bet.

• Probabilities are preferred for calculations; odds are common in gambling contexts.

Addition and Multiplication Rules

Addition Rule

• Intuitive Addition Rule: To find , add the number of ways A can occur and the number of ways B can occur, ensuring no outcome is counted twice.

• Formal Addition Rule:

• Disjoint (Mutually Exclusive) Events: Events that cannot occur at the same time; for such events, .

Multiplication Rule

• Intuitive Multiplication Rule: To find the probability that A occurs in one trial and B in another, multiply the probability of A by the probability of B, considering whether B is affected by A.

• Formal Multiplication Rule:

• Independent Events: Occurrence of one does not affect the probability of the other.

• Dependent Events: Occurrence of one affects the probability of the other.

• Sampling:

  • With replacement: selections are independent.

  • Without replacement: selections are dependent, unless the sample size is no more than 5% of the population (5% guideline).

Conditional Probability

• Conditional Probability: Probability of event B given that event A has occurred, denoted .

• Formula:

• Note: in general; confusing these is called "confusion of the inverse."

Finding the Probability of "At Least One"

• "At least one" means one or more occurrences.

• The complement is "none" of the event occurring.

• Formula:

Counting Principles

Multiplication Counting Rule

• If a sequence of events can occur in ways, the total number of outcomes is:

Factorial Rule

• The number of ways to arrange n different items (order matters):

• By definition, .

Permutations

• Permutations: Arrangements where order matters.

• Number of permutations of n items taken r at a time:

• When some items are identical: where are counts of identical items.

Combinations

• Combinations: Arrangements where order does not matter.

• Number of combinations of n items taken r at a time:

Mnemonic Devices

• "Permutations Position": Order matters in permutations.

• "Combinations Committee": Order does not matter in combinations.

Key Vocabulary

• Compound Event: Combines two or more simple events.

• Disjoint (Mutually Exclusive) Events: Cannot occur at the same time.

• Independent Events: Occurrence of one does not affect the other.

• Dependent Events: Occurrence of one affects the other.

• Conditional Probability: Probability with additional information.

• Prior Probability: Initial probability before new information.

• Posterior Probability: Revised probability after new information.

• Simulation: Process that mimics a real procedure to estimate probabilities.

Summary Table: Probability Approaches and Rules

Example Applications

• Example 1 (Classical Probability): What is the probability of drawing an ace from a standard deck of 52 cards?

  • There are 4 aces and 52 cards:

• Example 2 (Complement Rule): Probability of getting at least one head in three coin tosses.

  • Probability of no heads (all tails):

  • Probability of at least one head:

• Example 3 (Permutations): Number of ways to arrange 3 books on a shelf:

• Example 4 (Combinations): Number of ways to choose 2 students from 5:

Additional info: This guide expands on the original notes by providing full definitions, formulas, and examples for each probability and counting principle, ensuring a self-contained and comprehensive study resource for introductory statistics students.