MGU Mathematics Series — High School Statistics
Part III: Relationships, Models, and Algebraic Statistics
Sample Spaces · Events · Probability Rules · Simulations
Chapter 8 of 14
MGU Mathematics Series  ·  High School Statistics  ·  Library Textbook Edition

Chapter 8:
Probability
Foundations

Outcomes, sample spaces, events, experimental and theoretical probability, complements, unions, intersections, tree diagrams, area models, and simulations — the mathematics of structured uncertainty

■ Math Level: High School Statistics ■

■ 14 Sections  •  Part III of 5 ■

Follows Chapters 1–7 (Data, Graphs, Models)  ·  Prepares students for Chapter 9: Conditional Probability and Counting

Chapter 8 — Front Matter

Purpose and Overview

Up to this point in the course, statistics has focused on describing and modeling data that has already been collected. Probability shifts the focus forward: instead of asking "What happened in this data set?" it asks "How likely is it that something will happen?" This shift is important. Statistics ultimately exists to help people make decisions under uncertainty — and that requires a language for talking about uncertainty with precision. Probability provides that language.

In this chapter, students will study outcomes and sample spaces, define and classify events, calculate experimental and theoretical probabilities, apply the fundamental rules of probability, use tree diagrams and area models to organize complex situations, and use simulations to estimate probabilities that are difficult to calculate directly.

By the end of Chapter 8, students should understand that probability does not predict individual events with certainty. It describes long-run patterns and degrees of likelihood. A probability of 0.7 does not mean "this will happen 7 times out of the next 10." It means that over a very large number of repetitions, the event will occur approximately 70 percent of the time.

Opening Question

A weather forecast says there is a 70 percent chance of rain on Saturday. You have an outdoor picnic planned. Should you reschedule? Most people have heard statements like this hundreds of times, but relatively few have stopped to ask what "70 percent chance of rain" actually means. Understanding what probability means — and what it does not mean — is necessary for making good decisions about risk, evidence, and uncertainty in everyday life, science, medicine, finance, and public policy.

14Sections
13Objectives
18Key Terms
12+Worked Examples

Why This Chapter Matters

Probability appears across science (reliability of measurements), medicine (diagnostic likelihoods and treatment risks), finance and insurance (pricing policies, modeling risk), games (understanding fairness and expected outcomes), public policy (evaluating evidence from studies), and technology (machine learning, spam filters, recommendation algorithms).

Beyond specific applications, probability is foundational to the inference chapters ahead. Confidence intervals, hypothesis tests, p-values, and sampling distributions all depend directly on understanding what probability means and how it works. A student who enters Chapter 12 without a working understanding of probability will find statistical inference difficult to interpret. A student who enters with a strong probability foundation will find it much more accessible.

Learning Objectives

  1. 1Explain what probability measures and why it is described as structured uncertainty.
  2. 2Identify outcomes and construct sample spaces for simple and compound events.
  3. 3Define events and calculate their probabilities from sample spaces and tables.
  4. 4Calculate experimental probability from observed data and interpret its meaning.
  5. 5Calculate theoretical probability when all outcomes are equally likely.
  6. 6Use the complement rule to find the probability of an event not occurring.
  7. 7Identify unions and intersections of events and interpret them correctly.
  8. 8Apply the probability rules for complements, unions, and mutually exclusive events.
  9. 9Construct and use tree diagrams to organize and count outcomes.
  10. 10Interpret geometric area models as visual representations of probability.
  11. 11Design and run simple simulations to estimate probabilities.
  12. 12Identify common probability mistakes and explain why they occur.
  13. 13Apply probability reasoning to real-world situations in games, risk, and science.
Chapter 8 — Contents

Sections at a Glance

Sections 8.1 – 8.3
Measuring Uncertainty: Probability, Outcomes, and Events
What Probability Measures · Sample Spaces · Events and Their Probabilities
Section 8.1

What Probability Measures

Every day, people make decisions under uncertainty. "It will probably rain this afternoon." "That medicine is unlikely to cause serious side effects." These informal judgments are useful, but they are imprecise. Probability is the mathematical framework that makes judgments about uncertainty precise.

Formally, probability is a number between 0 and 1 assigned to an event. A probability of 0 means the event is impossible. A probability of 1 means the event is certain. A probability of 0.5 means the event occurs in about half of all repetitions over the long run.

Probability Range
0 ≤ P(A) ≤ 1   for any event A

The notation P(A) is read "the probability of event A." Probability describes long-run relative frequency, not the outcome of any single event. If you flip a fair coin, the probability of getting heads is 0.5. This does not mean that in two flips you will get exactly one head and one tail. It means that if you flip the coin a very large number of times, the proportion of heads will get closer and closer to 0.5.

ProbabilityInformal Meaning
0Impossible
Near 0 (e.g., 0.01)Extremely unlikely
0.25Unlikely but possible
0.50About as likely as not
0.75More likely than not
Near 1 (e.g., 0.99)Almost certain
1Certain
Student Warning

Probability describes long-run behavior over many repetitions. It does not guarantee what happens on the next trial or the next ten trials. A probability of 0.5 does not mean "it will happen exactly half the time in small samples."

Random Processes

Probability applies to random processes — situations where individual outcomes are unpredictable but long-run patterns are stable and describable. The word "random" in statistics simply means the outcome varies in a way that follows a probability model, even if individual results are not predictable.

Section 8.2

Outcomes and Sample Spaces

An outcome is a single possible result of a random process. The sample space is the complete set of all possible outcomes, typically labeled S. Outcomes must be mutually exclusive (no two can happen simultaneously in one trial) and exhaustive (all possibilities are covered).

Random ProcessSample Space S
Single coin flipS = {H, T}
Single standard die rollS = {1, 2, 3, 4, 5, 6}
Two coin flipsS = {HH, HT, TH, TT}
Equally Likely Outcomes

When all outcomes are equally likely, each outcome has the same probability. For a fair six-sided die, each face has probability 1/6. When outcomes are not equally likely, other methods — data, models, or formulas — are needed to assign probabilities.

Example 8.2.1
Two-stage sample space — die and coin
  • ProblemA fair four-sided die (numbered 1–4) is rolled, then a fair coin is flipped. Write the full sample space.
  • ThinkEach outcome is a pair: a die number and a coin result. Organize with a table to avoid missing any outcomes.
  • Represent
    1234
    H1H2H3H4H
    T1T2T3T4T
  • CalculateS = {1H, 2H, 3H, 4H, 1T, 2T, 3T, 4T} → 8 outcomes total (4 × 2 = 8)
  • Check4 × 2 = 8. The table confirms all outcomes are covered and each is unique.
  • InterpretBecause both the die and the coin are fair, all 8 outcomes are equally likely, each with probability 1/8.
Answer: S = {1H, 2H, 3H, 4H, 1T, 2T, 3T, 4T} — 8 equally likely outcomes.
Section 8.3

Events

An event is any collection of outcomes from the sample space. When all outcomes are equally likely, the probability of an event is:

Probability of an Event — Equally Likely Outcomes
P(A) = (number of outcomes in A) / (total number of outcomes in S)
Example 8.3.1
Finding the probability of an event
  • ProblemA bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. One marble is selected at random. What is the probability that it is blue?
  • ThinkAll 10 marbles are equally likely to be selected. The event of interest is "blue marble."
  • RepresentLet S = all 10 marbles, B = the 5 blue marbles.
  • CalculateP(B) = 5/10 = 1/2 = 0.5
  • Check5 blue marbles out of 10 total — exactly half of the sample space.
  • InterpretIf one marble is selected at random, there is a 50% chance it is blue. Over many repetitions, a blue marble would be selected about half the time.
Answer: P(blue) = 0.5
Student Warning

The formula P(A) = (outcomes in A) / |S| only works when all outcomes in the sample space are equally likely. If outcomes have different probabilities, simply counting outcomes will give incorrect results.

Sections 8.4 – 8.5
Experimental and Theoretical Probability
Observing vs. Calculating · Law of Large Numbers · Equally Likely Outcomes
Section 8.4

Experimental Probability

Theoretical probability requires knowing the structure of the sample space and assuming equally likely outcomes. But what about situations where the structure is unknown or outcomes are not equally likely? Experimental probability provides an estimate based on observed data:

Experimental Probability
P(event) ≈ (number of times the event occurred) / (total number of trials)
The Law of Large Numbers

As the number of trials grows, the experimental probability gets closer and closer to the theoretical probability. This means small samples can give misleading experimental probabilities. With 10 coin flips, getting 7 heads does not mean the coin is unfair — with small samples, wide variation is expected.

Example 8.4.1
Calculating experimental probability
  • ProblemA student rolls a six-sided die 60 times and records 12 threes. What is the experimental probability of rolling a three?
  • ThinkExperimental probability uses observed data, not the structure of the sample space.
  • CalculateP_exp(three) = 12/60 = 1/5 = 0.20
  • CheckTheoretical probability: 1/6 ≈ 0.167. The experimental result of 0.20 is somewhat higher, but with only 60 rolls, some deviation from theory is expected.
  • InterpretBased on these 60 rolls, a three appeared about 20% of the time. The small gap from 16.7% is consistent with normal random variation and does not suggest the die is unfair.
Answer: P_exp(three) = 0.20 (theoretical: ≈ 0.167)
Example 8.4.2
Experimental probability in quality control
  • ProblemA quality-control inspector at a factory examines 400 items and finds 12 defective. What is the experimental probability that a randomly selected item is defective?
  • RepresentLet D = the event that an item is defective. n = 400 items inspected.
  • CalculateP_exp(D) = 12/400 = 0.03 = 3%
  • Check0.03 means about 3 items in 100 — plausible for a manufacturing context.
  • InterpretApproximately 3% of items appear to be defective based on this sample. The factory might use this to estimate how many defective items to expect per production run, but the true defective rate for all items may differ — this is an estimate from 400 items.
Answer: P_exp(defective) = 0.03
Student Warning

A small number of trials can produce an experimental probability very different from the theoretical probability. This is expected, not alarming. Never conclude that a game is unfair, a coin is weighted, or a probability model is wrong based on a small number of observations.

Section 8.5

Theoretical Probability

When a random process has equally likely outcomes, the probability of any event can be calculated exactly from the sample space — no trials needed. This is theoretical probability. The key requirement is that all outcomes in the sample space are equally likely.

Example 8.5.1
Theoretical probability — sum of two four-sided dice
  • ProblemTwo fair four-sided dice (numbered 1–4) are rolled. What is the probability that the sum equals 5?
  • ThinkList all 4 × 4 = 16 equally likely outcomes using a table. Identify which sums equal 5.
  • Represent
    Die 1 \ Die 21234
    12345 ✓
    2345 ✓6
    345 ✓67
    45 ✓678
  • CalculateOutcomes with sum = 5: (1,4), (2,3), (3,2), (4,1) → 4 outcomes P(sum = 5) = 4/16 = 1/4 = 0.25
  • CheckSums range from 2 to 8. A sum of 5 is in the middle of this range and should have a reasonable probability. 0.25 is plausible.
  • InterpretWhen two fair four-sided dice are rolled, there is a 25% chance the sum equals 5.
Answer: P(sum = 5) = 0.25
Sections 8.6 – 8.9
Core Probability Rules
Complements · Unions and Intersections · Mutually Exclusive Events · Rules Summary
Section 8.6

Complements

Every event A has a complement, written Ac, which consists of all outcomes in the sample space that are not in A. Because every outcome either belongs to A or to Ac — not both, and not neither — the sample space is complete:

Complement Rule

P(A) + P(Aᶜ) = 1

P(Aᶜ) = 1 − P(A)

Use the complement when it is easier to calculate the probability an event does not occur and subtract from 1.

Example 8.6.1
Using the complement rule — multiples of 7
  • ProblemA box contains 50 tickets numbered 1–50. One ticket is drawn at random. What is the probability of drawing a ticket that is not a multiple of 7?
  • ThinkIt is easier to count multiples of 7 from 1 to 50 (a small set), then use the complement.
  • CalculateMultiples of 7 from 1–50: 7, 14, 21, 28, 35, 42, 49 → 7 outcomes P(multiple of 7) = 7/50 = 0.14 P(not a multiple of 7) = 1 − 0.14 = 0.86
  • Check0.14 + 0.86 = 1.00 ✓
  • InterpretThere is an 86% chance of drawing a number that is not a multiple of 7.
Answer: P(not a multiple of 7) = 0.86
Section 8.7

Unions and Intersections

Union and Intersection Definitions

A ∪ B — Union: all outcomes in A, in B, or in both. In plain language: "A or B (or both)."

A ∩ B — Intersection: all outcomes in both A and B. In plain language: "A and B together."

P(A ∪ B) = P(A) + P(B) − P(A ∩ B) — General Addition Rule

Venn Diagram

Draw a rectangle representing the full sample space S. Inside it, draw two overlapping circles: label the left circle A and the right circle B. The overlap (intersection) represents A ∩ B — outcomes in both. The combined area of both circles represents A ∪ B — outcomes in either or both. The region outside both circles represents outcomes in neither event.

Example 8.7.1
Union and intersection — hearts and face cards
  • ProblemOne card is drawn from a standard 52-card deck. Let A = drawing a heart. Let B = drawing a face card (jack, queen, or king of any suit). Find P(A ∪ B) and P(A ∩ B).
  • ThinkIdentify each event, then find their overlap. Don't count the three heart face cards twice.
  • RepresentA = hearts: 13 cards. B = face cards: 3 × 4 = 12 cards. A ∩ B = heart face cards: J♥, Q♥, K♥ = 3 cards.
  • CalculateP(A) = 13/52, P(B) = 12/52, P(A ∩ B) = 3/52 P(A ∪ B) = 13/52 + 12/52 − 3/52 = 22/52 = 11/26 ≈ 0.423
  • Check22 distinct cards are either hearts, face cards, or both. 13 + 12 − 3 = 22. ✓
  • InterpretIf one card is drawn at random, the probability that it is a heart or a face card (or both) is approximately 42.3%.
Answer: P(A ∩ B) = 3/52 ≈ 0.058; P(A ∪ B) = 22/52 ≈ 0.423
Section 8.8

Mutually Exclusive Events

Two events are mutually exclusive (also called disjoint) if they cannot both occur on the same trial: A ∩ B = ∅. When events are mutually exclusive, the General Addition Rule simplifies because there is nothing to subtract.

Addition Rule for Mutually Exclusive Events

If A ∩ B = ∅ (A and B cannot both occur):

P(A ∪ B) = P(A) + P(B)

Student Warning

Students often treat all events as mutually exclusive, leading to incorrect addition. Events can overlap. The events "rolling an even number" and "rolling a number greater than 4" are not mutually exclusive because rolling a 6 satisfies both. Always check whether the intersection is empty before using the simpler addition rule.

Example 8.8.1
Mutually exclusive events — colored marbles
  • ProblemOne marble is drawn from a bag with 3 red, 4 blue, and 5 green marbles. Let A = red, B = blue. Are A and B mutually exclusive? Find P(A ∪ B).
  • ThinkCan a single marble be both red and blue? No — a marble has exactly one color. So A and B are mutually exclusive.
  • Calculaten = 3 + 4 + 5 = 12 total marbles P(A) = 3/12 = 1/4, P(B) = 4/12 = 1/3 P(A ∪ B) = 3/12 + 4/12 = 7/12 ≈ 0.583
  • Check7 of 12 marbles are red or blue. 7/12 ≈ 0.583 ✓
  • InterpretIf one marble is drawn at random, there is approximately a 58.3% chance of drawing a red or blue marble.
Answer: Yes, mutually exclusive. P(red or blue) = 7/12 ≈ 0.583
Section 8.9

Probability Rules

The core probability rules gathered together make it easier to recognize which rule applies in a given situation. Before applying any rule, ask: Are outcomes equally likely? Are events mutually exclusive? Does the problem involve a complement, union, or intersection?

RuleFormulaWhen to Use
Probability Range0 ≤ P(A) ≤ 1Always — no probability can be negative or exceed 1
Sample SpaceP(S) = 1All outcomes together are certain
ComplementP(Aᶜ) = 1 − P(A)When "not A" is easier to calculate than A directly
General AdditionP(A ∪ B) = P(A) + P(B) − P(A ∩ B)Any two events, especially when they can overlap
Mutually Exclusive AdditionP(A ∪ B) = P(A) + P(B)Only when A and B cannot both occur
Equally Likely OutcomesP(A) = outcomes in A / |S|Only when all outcomes are equally likely
Example 8.9.1
Applying multiple rules — clubs and class membership
  • ProblemA class has 30 students. 18 are in a science club, 12 are in a math club, and 5 are in both clubs. One student is selected at random. (a) P(science); (b) P(at least one club); (c) P(neither club).
  • ThinkThe 5 students in both clubs mean science and math are not mutually exclusive — use the General Addition Rule for (b), then complement for (c).
  • Calculate(a) P(science) = 18/30 = 0.60 (b) P(science ∪ math) = 18/30 + 12/30 − 5/30 = 25/30 ≈ 0.833 (c) P(neither) = 1 − 25/30 = 5/30 ≈ 0.167
  • Check25 students are in at least one club; 5 are in neither. 25 + 5 = 30 ✓
  • InterpretThere is about an 83.3% chance the selected student is in at least one club, and a 16.7% chance in neither.
Answer: (a) 0.60; (b) ≈ 0.833; (c) ≈ 0.167
Sections 8.10 – 8.13
Tools: Tree Diagrams, Area Models, and Simulations
Tree Diagrams · Area Models · Simulations · Applications in Games, Risk, and Science
Section 8.10

Tree Diagrams

When a random process has two or more stages, tree diagrams provide a structured way to list and count all outcomes by branching at each stage. A tree starts from a single point. At each stage, branches extend to the right — one for each possible outcome. Reading any single path from left to right gives one complete outcome. To find the probability of any complete path, multiply the probabilities along the branches.

Example 8.10.1
Tree diagram for two coin flips
  • ProblemA fair coin is flipped twice. Draw a tree diagram. What is the probability of getting at least one head?
  • ThinkFirst flip: 2 outcomes. For each, second flip: 2 more. Total: 2 × 2 = 4 equally likely outcomes.
  • Represent
    H ──→ HH (prob 1/4) H ──┤ T ──→ HT (prob 1/4) Start ──┤ H ──→ TH (prob 1/4) T ──┤ T ──→ TT (prob 1/4)
  • Calculate"At least one head" includes HH, HT, TH → 3 of 4 outcomes P(at least one head) = 3/4 = 0.75
    Check via complement: P(no heads) = P(TT) = 1/4 P(at least one head) = 1 − 1/4 = 3/4 ✓
  • InterpretWhen a fair coin is flipped twice, there is a 75% chance of getting at least one head.
Answer: P(at least one head) = 3/4 = 0.75
Example 8.10.2
Tree diagram with unequal probabilities — spinner and coin
  • ProblemA spinner has three sections: Red (0.5), Blue (0.3), Green (0.2). The spinner is spun, then a fair coin is flipped. What is P(Red and Heads)?
  • ThinkOutcomes are not all equally likely (spinner has unequal sections). Multiply probabilities along the path.
  • CalculateP(Red and Heads) = 0.5 × 0.5 = 0.25
    All six paths check: R–H: 0.5×0.5 = 0.25 | R–T: 0.5×0.5 = 0.25 B–H: 0.3×0.5 = 0.15 | B–T: 0.3×0.5 = 0.15 G–H: 0.2×0.5 = 0.10 | G–T: 0.2×0.5 = 0.10 Sum = 1.00 ✓
  • InterpretThe probability of Red and Heads is 0.25, or 25%.
Answer: P(Red and Heads) = 0.25
Section 8.11

Area Models for Probability

Probability has a natural geometric interpretation: area. When the sample space is represented by a region with total area equal to 1, the probability of any event is proportional to the area of the region representing that event. An area model is usually a rectangle divided into sub-regions, each labeled with its event and probability. This connection between area and probability is part of the reason that normal distributions can be represented as curves where area under the curve equals probability.

Example 8.11.1
Area model — simple spinner
  • ProblemA spinner has four sections: Red (40%), Blue (25%), Green (20%), Yellow (15%). Represent with an area model and find P(Red or Green).
  • Represent
    Area Model Display

    Draw a rectangle of total area 1. Divide it into four vertical strips proportional to each probability: Red = 0.40 width, Blue = 0.25 width, Green = 0.20 width, Yellow = 0.15 width. Label each strip with its color and probability. Verify: 0.40 + 0.25 + 0.20 + 0.15 = 1.00.

  • CalculateRed and Green are mutually exclusive (spinner can't land on both). P(Red or Green) = 0.40 + 0.20 = 0.60
  • CheckRemaining area: 0.25 + 0.15 = 0.40 = complement. 0.60 + 0.40 = 1.00 ✓
  • InterpretThere is a 60% chance the spinner lands on Red or Green.
Answer: P(Red or Green) = 0.60
Example 8.11.2
Two-stage area model — grade level and textbook preference
  • Problem60% of students are in Grade 11 and 40% in Grade 12. Among Grade 11 students, 70% prefer digital textbooks. Among Grade 12 students, 50% prefer digital. Find the probability that a randomly selected student is in Grade 12 and prefers digital textbooks.
  • Represent
    PreferenceGrade 11 (width 0.60)Grade 12 (width 0.40)
    Digital0.60 × 0.70 = 0.420.40 × 0.50 = 0.20
    Not Digital0.60 × 0.30 = 0.180.40 × 0.50 = 0.20
  • CalculateP(Grade 12 and digital) = 0.40 × 0.50 = 0.20
  • CheckFour sub-regions: 0.42 + 0.18 + 0.20 + 0.20 = 1.00 ✓
  • InterpretThere is a 20% chance that a randomly selected student is in Grade 12 and prefers digital textbooks.
Answer: P(Grade 12 and digital) = 0.20
Section 8.12

Simulations

Some probabilities are difficult or impossible to calculate exactly using formulas. A simulation estimates probability by repeating a random process many times using a random device — a coin, dice, a random number table, or a calculator — and recording the proportion of times the event of interest occurs.

Steps for Designing a Simulation

1. Identify the random process and define the event of interest.

2. Choose a random device that matches the probabilities in the problem.

3. Define a trial — specify what one repetition looks like.

4. Run many trials (100, 500, or 1000+ for reliable estimates).

5. Record and calculate — count how many trials produce the event and divide by total trials.

6. Interpret — the proportion of successful trials estimates P(event).

Example 8.12.1
Designing a simulation — basketball free throws
  • ProblemA basketball player makes 70% of free throws. Design a simulation to estimate P(at least 4 makes in 5 shots).
  • ThinkP(make) = 0.70. Use random digits 0–9 where digits 0–6 represent a made free throw (70%) and digits 7–9 represent a miss (30%).
  • RepresentOne trial = five random digits (five free throws). Count digits in range 0–6 as makes.
  • Calculate
    TrialDigitsMakes (0–6)Count≥4?
    13 5 9 1 0M M – M M4Yes
    27 2 4 6 8– M M M –3No
    31 0 5 3 2M M M M M5Yes
    49 9 7 4 1– – – M M2No
    56 3 0 7 5M M M – M4Yes
    In these 5 trial runs: event occurred 3 times P_sim ≈ 3/5 = 0.60 (unreliable with only 5 trials)
  • CheckTheoretical value (using binomial methods from Chapter 10) ≈ 0.528. With many more trials, the simulation estimate converges toward this value.
  • InterpretFive trials cannot provide a reliable estimate — run at least 100 trials for a stable result. The method is correct; the sample is too small.
Answer: Simulation estimate from 5 trials = 0.60 (needs many more trials to be reliable; theoretical ≈ 0.528)
Student Warning

Simulations produce estimates, not exact answers. The accuracy depends critically on the number of trials. Five or ten trials cannot provide a trustworthy estimate. Run at least 100 trials, and ideally 1000 or more, for a reliable result.

MGU Library Connection
  • Simulations and the Law of Large Numbers are foundational to bootstrap methods and Monte Carlo simulations — core topics in Data Science Foundations.
  • The intuition that more trials give better estimates is the same principle that makes large samples more reliable throughout statistical inference (Chapters 12–13).
  • Probability Rules Reference — Appendix H of this textbook
  • Calculator and Spreadsheet Skills for simulations — Appendix O of this textbook
Section 8.13

Probability in Games, Risk, and Science

Games and Fairness

Probability is the foundation of game design, gambling, and decision-making under risk. When a game is fair, each player has an equal expected outcome over the long run. Consider a carnival game where a player rolls a fair die — winning $5 if it shows 6 and losing $1 otherwise:

Fairness Check

P(win) = 1/6, P(lose) = 5/6. Average result per play: (1/6)(5) + (5/6)(−1) = 5/6 − 5/6 = 0. An average result of 0 means the game is mathematically fair on average — neither player has a long-run advantage. Expected value is studied fully in Chapter 10.

Risk and Insurance

Insurance companies use probability to set prices. If the probability of a house fire in a given year is 0.002 and the average claim is $150,000, the company needs to collect more than 0.002 × $150,000 = $300 per policy per year just to break even on claims. Individual customers accept this trade-off because they are exchanging a certain small loss (the premium) for protection against a large uncertain loss.

Data Ethics Note

Scientists, journalists, and policymakers who misinterpret probability can reach incorrect conclusions with real consequences. Claiming a treatment "definitely works" based on a small study, or dismissing a risk as negligible based on one unlikely outcome, are both mistakes rooted in misunderstanding probability. Statistical literacy protects against these errors.

Example 8.13.1
Probability in a medical context — rapid strep test
  • ProblemA rapid strep test correctly identifies strep infection 90% of the time when present and gives a correct negative 95% of the time when no infection is present. In a clinic where 10% of patients have strep, what is the probability that a randomly selected patient tests positive?
  • ThinkTwo ways to test positive: (1) have strep and test correctly positive, or (2) no strep but get a false positive. These paths are mutually exclusive.
  • Represent
    Tree Diagram

    From the start, branch into Strep (0.10) and No Strep (0.90). From Strep: branch to Positive (0.90) and Negative (0.10). From No Strep: branch to Positive (false positive, 0.05) and Negative (0.95).

  • CalculateP(positive) = P(strep AND positive) + P(no strep AND positive) = 0.10 × 0.90 + 0.90 × 0.05 = 0.09 + 0.045 = 0.135
  • CheckP(negative) = 0.10×0.10 + 0.90×0.95 = 0.01 + 0.855 = 0.865. Sum: 0.135 + 0.865 = 1.00 ✓
  • InterpretApproximately 13.5% of patients will test positive — including 9% true positives and 4.5% false positives. A positive test does not guarantee infection, especially when the disease is relatively rare. This kind of reasoning is important in medicine and public health.
Answer: P(positive test) = 0.135 (includes 0.09 true positive + 0.045 false positive)
Section 8.14
Common Probability Mistakes
Gambler's Fallacy · Overlapping Events · Misreading "or" · Probability vs. Certainty
Section 8.14

Common Probability Mistakes

Mistake 1 — The Gambler's Fallacy
"I flipped heads 5 times in a row. Tails is overdue — the probability of tails is now very high."

A fair coin has no memory. The probability of tails on the next flip is always 0.5, regardless of what happened before. Past outcomes in a random process do not influence future outcomes. The Gambler's Fallacy causes people to make decisions based on a false belief that random patterns will "even out" in the short run.

Mistake 2 — Treating Non-Mutually-Exclusive Events as Mutually Exclusive
P(A ∪ B) = P(A) + P(B) (applied to all events without checking)

This formula only applies when A and B cannot both occur. If events overlap, simply adding their probabilities double-counts the intersection. Always check whether A ∩ B = ∅ before using the simplified addition rule. Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B) for any two events.

Mistake 3 — Confusing "and" with "or"
Computing P(A and B) when the problem asks for P(A or B), or vice versa.

In mathematics, P(A ∪ B) means "A or B or both" (inclusive or). In everyday English, "or" sometimes excludes both. Read the problem carefully. Union (∪) is "or." Intersection (∩) is "and." They are very different events.

Mistake 4 — Treating Probability as a Guarantee or Impossibility
"The probability is 0.9, so it will definitely happen." Or: "The probability is 0.01 — it can't happen."

A probability of 0.9 means the event occurs about 90% of the time over many repetitions — not that it is certain on the next trial. A probability of 0.01 means unlikely, not impossible. Unlikely events occur. Probability quantifies degree of likelihood; it does not guarantee or rule out individual outcomes.

Mistake 5 — Applying the Equally Likely Formula When Outcomes Are Not Equally Likely
"There are two outcomes — car accident or no accident — so each has probability 0.5."

The formula P(A) = (outcomes in A) / |S| only applies when all outcomes in the sample space are equally likely. Two outcomes do not automatically mean each has probability 0.5. Always ask: are all outcomes equally likely? If not, probabilities must come from data or models.

Mistake 6 — Concluding from Small Samples
"I rolled the die 10 times and got a 3 only once. The probability of rolling a 3 must be less than 1/6."

Experimental probability from a small sample may differ significantly from theoretical probability. This is expected — it is not evidence that the die is unfair or the model is wrong. The Law of Large Numbers requires many, many trials before the experimental probability reliably approximates the theoretical one.

Chapter 8 — Back Matter

Chapter Summary

Key Terms Review

probability
A number between 0 and 1 describing how likely an event is to occur over many repetitions of a random process.
outcome
A single possible result of a random process.
sample space
The complete set of all possible outcomes of a random process, labeled S.
event
Any collection of outcomes from a sample space. May consist of one, several, or all outcomes.
simple event
An event consisting of exactly one outcome from the sample space.
complement
The complement of event A, written Aᶜ, is all outcomes in S that are not in A. P(Aᶜ) = 1 − P(A).
experimental probability
Probability estimated from observed data: (event occurrences) / (total trials). Approaches theoretical probability as trials increase.
theoretical probability
Probability calculated from the structure of the sample space when all outcomes are equally likely.
union (A ∪ B)
All outcomes in A, in B, or in both. "A or B." P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
intersection (A ∩ B)
All outcomes in both A and B simultaneously. "A and B." If mutually exclusive, P(A ∩ B) = 0.
mutually exclusive
Two events that cannot both occur on the same trial. Their intersection is empty: A ∩ B = ∅.
Law of Large Numbers
As the number of trials of a random process increases, experimental probability approaches theoretical probability.
tree diagram
A branching diagram listing all outcomes of a multi-stage random process. Multiply probabilities along branches to find path probabilities.
area model
A geometric representation of probability using regions whose areas equal the probabilities of events. Total area = 1.
simulation
A method of estimating probability by using a random device to imitate a real random process many times.
Venn diagram
A rectangle (sample space) with overlapping circles (events) showing unions, intersections, and complements geometrically.
Gambler's Fallacy
The mistaken belief that a random process has memory — that past outcomes influence future probabilities for independent events.
random process
A situation where individual outcomes are unpredictable but long-run patterns are stable and describable by a probability model.

Concept Review Questions

Answer the following in complete sentences.

  1. What does a probability of 0.5 mean for a random process? Does it guarantee any specific short-run outcome?
  2. What is the difference between experimental probability and theoretical probability? When is each more useful?
  3. Explain in your own words what the Law of Large Numbers says. Why does it matter for experimental probability?
  4. What does it mean for two events to be mutually exclusive? Give one example of mutually exclusive events and one example of events that are not.
  5. Why is P(A ∪ B) = P(A) + P(B) potentially incorrect if used without checking for mutual exclusivity?
  6. Describe the purpose of a tree diagram. In what situations is a tree diagram especially useful?
  7. How does an area model represent probability geometrically?
  8. Why do simulations require a large number of trials to be reliable?
  9. What is the Gambler's Fallacy? Explain why it is incorrect.
  10. A student says, "Either I will pass this test or I won't, so the probability of passing is 0.5." What mistake is this student making?

Sample Space Practice

  1. A spinner has five equal sections: A, B, C, D, E. Write the sample space.
  2. A student randomly selects a day of the week for a meeting. Write the sample space.
  3. Two fair coins are flipped. Write the full sample space for the pair of outcomes.
  4. A fair four-sided die (numbered 1–4) is rolled twice. Write the full sample space using ordered pairs.
  5. A bag contains one red ball (R), one blue ball (B), and one yellow ball (Y). One ball is drawn, recorded, and replaced. A second ball is drawn. Write the sample space.
  6. A student chooses a main course (pasta or chicken) and a drink (water, juice, or milk). Write the sample space of meal combinations. How many equally likely outcomes are there? What is the probability of choosing pasta with juice?
PracticeProbability Rule Practice
  1. P(A) = 0.35. Find P(Aᶜ).
  2. P(B) = 3/8. Find P(Bᶜ).
  3. P(A)=0.4, P(B)=0.3, P(A∩B)=0.1. Find P(A∪B).
  4. P(A)=0.6, P(B)=0.25, A and B mutually exclusive. Find P(A∪B).
  5. P(A∪B)=0.7, P(A)=0.5, P(B)=0.4. Find P(A∩B).
  6. P(rolling a 4 on a fair die)?
  7. P(not rolling a 4)?
  8. P(rolling an odd number)?
  9. P(number < 3 or > 4)?
  10. P(drawing a king from 52 cards)?
  11. P(drawing a red card)?
  12. P(red king)?
  13. P(red card or king)?
  14. Survey of 200 students: 80 play sports, 70 play instrument, 30 do both. Find P(sports or instrument) and P(neither).
PracticeTree Diagram Practice
  1. A student randomly picks a shirt color (red, blue) and pants color (black, gray, tan). Draw a tree diagram and list the full sample space of outfits. How many outcomes? What is P(red shirt with gray pants)?
  2. A bag has 3 red and 2 blue marbles. One marble is drawn, recorded, and replaced. A second marble is drawn. Draw a tree diagram. What is the probability of drawing two red marbles?
  3. A spinner has three sections: P1 (0.5), P2 (0.3), P3 (0.2). The spinner is spun twice. Draw a tree diagram. What is P(P1 on both spins)?
  4. 60% of residents own a dog and 40% do not. Of dog owners, 70% also own a cat. Of non-dog-owners, 30% own a cat. Draw a tree diagram and find P(owns both a dog and a cat).
  5. Extend Problem 34: what is the probability that a resident owns at least one pet (dog, cat, or both)?
PracticeSimulation Problems
  1. A soccer player scores on penalty kicks 65% of the time. (a) Describe how to use random digits 0–9 to simulate one kick. (b) Define a trial of 5 penalty kicks. (c) Identify the outcome to count for P(at least 4 of 5 made). (d) Explain why 10 trials gives a poor estimate while 500 gives a better one.
  2. A multiple-choice test has 5 questions, each with 4 choices. A student guesses randomly. (a) Describe a simulation using a random device for one question. (b) Describe simulating the full 5-question test. (c) Identify the success outcome for P(3 or more correct by guessing).
  3. A bakery finds 80% of pastry customers also buy a beverage. Design a simulation to estimate P(exactly 5 of 6 customers buy a beverage). Include: device, trial definition, success outcome, and a plan for at least 50 trials.

Applications

  1. Weather. A meteorologist reports a 65% chance of rain on each of Saturday and Sunday, with independent days. Find P(rain both days), P(no rain either day), and P(rain on exactly one day).
  2. Quality Control. A production line makes chips; 2% are defective. Three chips are randomly selected. (a) Find P(all three non-defective). (b) Find P(at least one defective) using the complement. (c) Why might a factory care more about (b) than (a)?
  3. Medical Testing. A disease affects 1% of the population. A diagnostic test has 98% sensitivity and 97% specificity. (a) Draw a tree diagram. (b) Find P(testing positive). (c) Of all who test positive, what fraction actually has the disease? (d) What does this suggest about interpreting a positive test when disease is rare?
  4. Games. A game board spinner: Win (35%), Lose (65%). A player spins twice. (a) P(win twice). (b) P(win at least once). (c) P(never win). (d) Verify that answers to (b) and (c) sum to 1.

Error Analysis

In each problem, a student made a probability error. Identify the error and explain the correct approach.

Problem 43
"I flipped a coin 8 times and got heads each time. The probability of tails on the next flip must be very high because tails is 'overdue.'"

What is wrong with this reasoning? Name the fallacy and explain the correct interpretation.

Problem 44
P(rolling a 2 or an even number) = P(2) + P(even) = 1/6 + 3/6 = 4/6

What error was made? What is the correct probability?

Problem 45
"P(junior) = 1/4 and P(female) = 1/2, so P(junior or female) = 1/4 + 1/2 = 3/4."

Without additional information, why is this conclusion potentially incorrect? What additional information is needed?

Problem 46
Simulation design: flip a coin once. Heads = pass the 5-question quiz. Tails = fail.

A student uses this simulation to estimate the probability of passing a true/false quiz (5 questions, P = 0.5 each). What is wrong with this design?

Problem 47
Sample space for sum of two dice: {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Each sum is equally likely.

What is wrong with this sample space? How many equally likely outcomes are there really?

Chapter 8 Checkpoint

Chapter 8 Checkpoint
Complete all problems without notes — assesses all key skills and concepts from Chapter 8
Part A — Vocabulary and Concepts
  1. Define probability in your own words. What values can a probability take?
  2. What is the difference between an outcome and an event?
  3. State the Complement Rule and explain when you would use it.
  4. When is it valid to use the simplified addition rule P(A ∪ B) = P(A) + P(B)?
  5. In your own words, what does the Law of Large Numbers say about experimental probability?
Part B — Calculations
  1. A bag contains 6 white, 4 black, and 2 orange balls. One ball is drawn. (a) P(orange). (b) P(not white). (c) P(white or black).
  2. P(A) = 0.55, P(B) = 0.40, P(A ∩ B) = 0.20. (a) Find P(A ∪ B). (b) Find P(Aᶜ). (c) Are A and B mutually exclusive? Explain.
  3. A fair coin is flipped three times. Draw a tree diagram. Find: (a) P(exactly 2 heads). (b) P(at least 1 tail). (c) P(all same — all heads or all tails).
Part C — Interpretation and Application
  1. A game spinner: Yellow (0.45), Red (0.35), Blue (0.20). (a) Verify the probabilities sum to 1. (b) Find P(not yellow). (c) A player wins on Red or Blue. Find P(win).
  2. A survey of 300 adults: 120 exercise daily, 90 take vitamins daily, 40 do both. (a) P(exercises or takes vitamins). (b) P(neither). (c) P(exercises but does not take vitamins).
  3. A clinic: 15% of patients have the flu. A rapid flu test has 88% sensitivity and 92% specificity. (a) Draw a tree diagram. (b) Find P(patient tests positive). (c) Explain what a false positive result means for the patient.
Part D — Simulation
  1. A baseball player has a batting average of 0.300. (a) How would you use random digits to simulate one at-bat? (b) Describe how to simulate 4 at-bats. (c) Identify the event to track for P(at least 2 hits in 4 at-bats). (d) Why does increasing the number of simulated series improve the estimate?

Answers may be placed in the answer key.

Bridge Note — Probability and Statistical Inference
Chapter 9: Conditional Probability and Counting · and Beyond

Probability is not merely one topic among many in statistics — it is the language that makes inference possible. Every concept in Chapters 12 and 13 depends on probability reasoning: confidence intervals are built around the idea that sample statistics vary according to a probability distribution; a p-value is a probability; hypothesis testing is a formal decision-making process that uses probability to weigh evidence.

Chapter 9 extends probability to conditional probability — how probability changes when additional information is available. Conditional probability is essential for reasoning about dependence, independence, two-way tables, testing, and counting complex outcomes. The medical testing example in Section 8.13 showed a preview of this reasoning; Chapter 9 formalizes it completely.

Beyond this textbook, probability is central to college statistics, data science (probability models for classification and prediction), finance and economics (expected value, risk modeling, portfolio theory), medicine and public health (diagnostic testing, clinical trials), and physics, chemistry, and biology (quantum probability, thermodynamics, genetics). The habit of thinking carefully about uncertainty, sample spaces, evidence, and long-run patterns — established in this chapter — will serve students across every field that deals with data, risk, or decision-making.

MGU Library Connection
  • Probability Rules Reference — Appendix H of this textbook
  • Counting Formulas Reference — Appendix I (used in Chapter 9)
  • Calculator and Spreadsheet Skills for simulations — Appendix O of this textbook
  • Geometry, Chapter 10: Area and the Coordinate Plane (area models for probability connect geometry to statistics)
  • Algebra I, Chapter 8: Probability Foundations (earlier treatment at Algebra I level)
  • Data Science Foundations: Monte Carlo simulations and bootstrap methods
  • MGU Dictionary entries: probability, sample space, event, complement, union, intersection, mutually exclusive, Law of Large Numbers, simulation