Probability: Foundations and Approaches

Definitions and Notation

• Probability (P) quantifies the likelihood of an event occurring, expressed as a number between 0 and 1 (inclusive).

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

• Simple Event is an outcome that cannot be broken down further.

• Sample Space is the set of all possible simple events for a procedure.

• Notation: P(A) denotes the 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.

• Classical Approach (Equally Likely Outcomes):

• Subjective Probability: Estimated using knowledge of relevant circumstances.

• Simulation: When other approaches are not feasible, simulate the procedure to estimate probabilities.

Law of Large Numbers

• As a procedure is repeated many times, the relative frequency probability of an event approaches the actual probability.

• Probability estimates based on few trials can be inaccurate; more trials yield better approximations.

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.

Probability Properties and Cautions

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

• Do not assume outcomes are equally likely without justification.

• Avoid dividing numbers without understanding their meaning in context.

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

Significance in Probability

• A result is considered significantly high if the probability of observing x or more successes is ≤ 0.05.

• A result is significantly low if the probability of x or fewer successes is ≤ 0.05.

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

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

• Odds are commonly used in gambling; probabilities are preferred in scientific contexts.

Rules of Probability: Addition and Multiplication

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 together; 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: where is the probability of B given A has occurred.

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

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

Sampling and Independence

• Sampling with Replacement: Selections are independent.

• Sampling without Replacement: Selections are dependent.

• 5% Guideline: When the sample size is no more than 5% of the population, treat dependent events as independent for simplicity.

Complementary Events

• The probability of "at least one" occurrence is:

Conditional Probability

• Conditional Probability is the probability of event B occurring given that event A has already occurred. Notation:

• Formal Definition:

• In general, ; confusing these is called "confusion of the inverse."

Counting Rules: Fundamental Principles

Multiplication Counting Rule

• For a sequence of events, if the first can occur in ways, the second in ways, etc., the total number of outcomes is:

Factorial Rule

• The number of ways to arrange n different items (order matters): By definition, .

Permutations

• Permutations are 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 are selections 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 and Concepts

• 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 probability of the other.

• Conditional Probability: Probability with additional information.

• Prior Probability: Initial probability before new information.

• Posterior Probability: Revised probability after new information.

Summary Table: Probability Approaches and Counting Rules

Example: Calculating a Probability Using the Classical Approach

• Suppose a fair six-sided die is rolled. What is the probability of rolling a 4?

• There are 6 equally likely outcomes (1, 2, 3, 4, 5, 6). Only one outcome is a 4.

• Using the classical approach:

Example: Using the Multiplication Rule

• What is the probability of flipping two coins and both landing heads?

• Each coin flip is independent, .

Example: Counting Permutations

• How many ways can 3 books be arranged on a shelf?

• ways

Example: Counting Combinations

• How many ways can a committee of 2 be chosen from 4 people?

• ways

Additional info: This guide covers probability rules, significance, odds, and counting principles, providing a foundation for further study in probability and statistics.