The Mathematics of Maybe: Introduction to Probability

1. Chapter at a glance

• Probability measures the likelihood of an event occurring and is expressed on a scale from 0 (impossible) to 1 (certain).
• Random experiments involve chance: all possible outcomes are known, but the exact outcome cannot be predicted in advance.
• Experimental probability is obtained from actual trials or data as relative frequency: number of times the event occurred divided by total number of trials.
• Theoretical probability assumes all outcomes are equally likely and is calculated as number of favourable outcomes divided by total number of possible outcomes.
• The sample space S is the set of all possible outcomes of a random experiment; each outcome is an element of S and n(S) denotes the sample size.
• An event is any single outcome or combination of outcomes; it is a subset of the sample space.
• Tree diagrams visually list all outcomes of multi-step experiments by showing branches for each successive trial.
• As the number of trials increases, experimental probability approaches theoretical probability (Law of Large Numbers); each trial remains independent.

2. Definitions, theorems and results

• Random experiment: An experiment that can be repeated, where every repetition may give a different result and the outcome cannot be known in advance.
• Sample space (S): The set of all possible outcomes of a random experiment, listed within brackets and separated by commas; each outcome is an element of S; n(S) is the number of elements (sample size). The sample space must include every possible outcome exactly once.
• Event: Any single possible result or combination of results; it is a subset of the sample space.
• Experimental probability: Number of times the event occurred / Total number of trials. Also called relative frequency when obtained from data.
• Theoretical probability P(Event): Number of favourable outcomes / Number of possible outcomes, assuming all outcomes are equally likely.
• Probability scale: Values lie between 0 (impossible) and 1 (certain); values strictly between 0 and 1 indicate varying degrees of likelihood.
• No theorems, lemmas or axioms requiring proof are stated in the chapter. All results are definitional or follow directly from counting equally likely outcomes.

3. Formula sheet

4. Solved-example patterns

• Listing sample space: Identify the experiment, list every distinct possible outcome exactly once inside braces (e.g., coin toss → {H, T}; two coins → {HH, HT, TH, TT}).
• Experimental probability from data/trials: Count occurrences of the event in the given trials or table, divide by total trials or sample size; express as fraction or decimal.
• Theoretical probability (single step): Count favourable outcomes from the described sample space, divide by total possible outcomes; simplify fraction.
• Probability from word or letter data: Count letters or items satisfying the condition for favourable outcomes, divide by total letters/items.
• Using statistical sample data: Treat given frequencies as favourable, total sample size as denominator; scale estimate to larger population if required.
• Tree diagram construction: Draw branches for first trial, then split each branch for subsequent independent trials; list all terminal paths to obtain sample space.
• Event probability via tree or listing: Identify paths/outcomes belonging to the event, count them, divide by total paths/outcomes.

5. Common mistakes and exam pitfalls

• Treating outcomes as not equally likely when the problem states “fair” or “standard” die/coin (must assume equally likely unless stated otherwise).
• Omitting outcomes or duplicating them when writing sample space S.
• Using experimental probability formula when the problem asks for theoretical probability (or vice versa).
• Forgetting that each trial is independent; applying Gambler’s Fallacy (past results do not change next probability).
• Miscounting favourable outcomes when the event is “at least one”, “not red”, “greater than”, etc.
• Scaling sample results to population without using the given sample proportion correctly.
• Writing probability outside [0,1] or forgetting to express answers as simplified fractions/decimals/percents as required.