Probability Fundamentals

Understanding Probability

Probability is a measure of how likely an event is to occur, expressed as a number between 0 and 1. Statisticians use probability to evaluate explanations and make decisions based on data.

• Probability Notation: P denotes probability; A, B, C denote specific events. P(A) is the probability of event A occurring.

• Relative Frequency Approximation: Probability can be estimated by repeating a procedure and calculating:

• Classical Approach: If all outcomes are equally likely, and event A can occur in s ways out of n possible outcomes:

• Subjective Probability: Estimated using knowledge of relevant circumstances.

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

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.

• Do not calculate probability by simply dividing a smaller number by a larger one without understanding the context.

Complementary Events

• The complement of event A (denoted as A̅) consists of all outcomes where A does not occur.

• Relationship:

Significant 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; sometimes 0.01 or other values are used.

Probability Review

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

• Notation: P(A) is the probability of event A; P(\overline{A}) is the probability that A does not occur.

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.

Rules of Probability

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:

• "Or" suggests addition; avoid double-counting outcomes.

• 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:

• Independent Events: Occurrence of one does not affect 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 is less than 5% of the population (5% guideline).

Conditional Probability

• Notation: is the probability of B given A has occurred.

• Formula:

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

At Least Once Probability

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

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

• To find the probability of at least one occurrence:

Counting Rules

Multiplication Counting Rule

• If the first event can occur in ways, the second 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:

• Permutations with Identical Items: If there are alike, alike, ..., alike among n items:

Combinations

• Combinations: Arrangements where order does not matter.

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

• Mnemonic: "Permutations Position" (order matters), "Combinations Committee" (order does not matter).

Key Vocabulary

• Event: Any collection of results or outcomes of a procedure.

• Simple Event: An outcome that cannot be broken down further.

• Sample Space: All possible simple events.

• Compound Event: Combines two or more simple events.

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

• Independent Events: One event does not affect the probability of the other.

• Dependent Events: One event affects the probability of the other.

• Conditional Probability: Probability calculated with additional information.

• Prior Probability: Initial probability before new information.

• Posterior Probability: Probability revised with new information.

• Permutation: Arrangement where order matters.

• Combination: Arrangement where order does not matter.

Summary Table: Probability Approaches and Counting Rules

Example: Calculating Probability

• Example 1: If a fair die is rolled, what is the probability of rolling a 4?

• There are 6 equally likely outcomes; only one is a 4.

• Example 2: How many ways can 3 books be arranged on a shelf?

• Order matters, so use the factorial rule:

• Example 3: How many ways can a committee of 2 be chosen from 4 people?

• Order does not matter, so use combinations:

Additional info: Examples and table entries were added for clarity and completeness.