MGU Mathematics Series — Algebra II
Part V: Geometry Connections, Trigonometry, Probability, and Modeling
Probability · Counting · Distributions · Regression · Decision-Making
Chapter 14 of 14
MGU Mathematics Series  ·  Algebra II  ·  Library Textbook Edition

Chapter 14:
Probability, Statistics,
and Algebraic Decision-Making

Sample spaces, conditional probability, permutations, combinations, binomial experiments, normal distributions, regression models, and evidence-based reasoning — the mathematics of uncertainty

■ Math Level: Algebra II ■

■ 15 Sections  •  Part V of 5 ■

Closes Algebra II  ·  Prepares students for Precalculus, Trigonometry, Statistics, and Applied MGU Courses

Chapter 14 — Front Matter

Purpose and Overview

Mathematics is not only the study of certain answers. It is also the study of how to reason carefully when outcomes are uncertain, when data is incomplete, and when decisions must be made without perfect information. This chapter connects the algebraic tools built throughout this course to one of the most practical branches of mathematics: probability and statistics.

Algebra gives us models — equations, functions, and formulas that describe how quantities are related. Probability gives us tools for measuring uncertainty. Statistics gives us tools for understanding data, identifying patterns, and drawing conclusions. Together, these three areas of mathematics form the foundation of evidence-based reasoning in science, medicine, engineering, business, social science, and everyday life.

Chapter 14 closes Algebra II by showing that everything learned in this course — functions, equations, expressions, graphing, modeling — can be turned toward the question of understanding what is likely, what the data says, and what decisions are supported by the evidence.

Opening Question

A pharmaceutical company tests a new medication on two groups of patients. In Group A, 68 out of 100 patients show improvement. In Group B, 52 out of 100 patients show improvement. A reporter says the new drug "clearly works better." A statistician says the result "may be due to chance." Who is right? To answer this question, you need probability, knowledge of sample size, an understanding of distributions, and the ability to judge when differences are meaningful. By the end of this chapter, you will have the tools to think carefully about exactly these kinds of questions.

15Sections
14Objectives
30+Key Terms
12+Worked Examples

Learning Objectives

  1. 1Describe sample spaces and events using set language and organized lists.
  2. 2Calculate theoretical and experimental probability.
  3. 3Distinguish between independent and dependent events and apply the multiplication rules.
  4. 4Calculate conditional probability using formal notation and reasoning.
  5. 5Count ordered arrangements using permutation formulas.
  6. 6Count unordered selections using combination formulas.
  7. 7Recognize binomial situations and apply the binomial probability formula.
  8. 8Describe data distributions using shape, center, and spread.
  9. 9Interpret the normal distribution and use the empirical rule.
  10. 10Fit regression models to data and interpret slope and intercept in context.
  11. 11Compare linear, quadratic, exponential, and logarithmic models using residuals and fit.
  12. 12Make data-informed decisions and communicate conclusions clearly.
  13. 13Recognize and correct common errors in probability and statistical reasoning.
  14. 14Connect Algebra II ideas to Precalculus, Statistics, and applied mathematics.
Chapter 14 — Contents

Sections at a Glance

Sections 14.1 – 14.5
Probability: Measuring Uncertainty
Why It Matters · Sample Spaces · Theoretical Probability · Independence · Conditional Probability
Section 14.1

Why Probability and Statistics Matter

Mathematics typically asks questions with definite answers: What is the slope of this line? What are the zeros of this function? These questions assume a single correct answer waiting to be found. But many important questions in life do not have single correct answers — they have answers that depend on chance, on the behavior of many individuals, or on incomplete information. How likely is it to rain tomorrow? What is the chance that a manufactured part will fail? How confident can we be that a new drug is effective?

To answer questions like these, we need two complementary tools: probability, which measures the likelihood of outcomes, and statistics, which organizes and interprets data from observations. Probability tells us what to expect in theory. Statistics tells us what actually happened in practice — and whether what happened matches what theory predicts.

Algebra II builds the function and modeling skills that probability and statistics depend on. Exponential models appear in population growth; polynomial and rational models appear in regression; linear models appear constantly in data analysis. The work of this chapter will draw directly on the functions, equations, and graphs studied throughout the course.

Reading Data Critically

Not all claims made with numbers are correct. Understanding the difference between a result due to chance and one reflecting a real pattern, between correlation and causation, and between a well-chosen model and a misleading one — these distinctions matter enormously. As you move through this chapter, keep asking: What does this number really mean? How confident can I be in this conclusion? What could go wrong with this reasoning?

Section 14.2

Sample Spaces and Events

Every probability problem begins with an experiment — a process whose result is uncertain. Each possible result is called an outcome. The collection of all possible outcomes is the sample space, usually denoted \( S \).

ExperimentSample SpaceSize of S
Flip one coin\( \{H,\ T\} \)2
Roll one die\( \{1,\ 2,\ 3,\ 4,\ 5,\ 6\} \)6
Flip a coin and roll a die\( \{(H,1),\ (H,2),\ \ldots,\ (T,6)\} \)12
Draw one card from a standard deckAll 52 cards52

An event is any subset of the sample space. The complement of an event \( E \), written \( E^c \), contains all outcomes not in \( E \). Two important set operations: the union \( A \cup B \) contains outcomes in \( A \) or \( B \) or both; the intersection \( A \cap B \) contains only outcomes in both. Events are mutually exclusive if \( A \cap B = \emptyset \) — they cannot occur simultaneously.

Example 14.2.1
Listing a sample space and its complement
  • ProblemA card is drawn from a standard 52-card deck. Define the event "draw a face card" (Jack, Queen, or King). List the event size and complement size.
  • ThinkA standard deck has 4 suits. Each suit has one Jack, one Queen, and one King, giving \( 4 \times 3 = 12 \) face cards.
  • Solve |E| = 12 (face cards: J, Q, K across 4 suits) |E^c| = 52 - 12 = 40 (all non-face cards)
  • Check\( 12 + 40 = 52 \). ✓
Answer: 12 face cards in the event; 40 cards in the complement.
Section 14.3

Theoretical and Experimental Probability

Theoretical Probability

When all outcomes are equally likely:

\[ P(E) = \frac{\text{number of outcomes in } E}{\text{total number of outcomes in } S} \]

P(E) is always between 0 and 1.

P(S) = 1.    P(Ø) = 0.    P(E^c) = 1 - P(E).

Experimental probability (relative frequency) is calculated from actual data: divide the number of times the event occurred by the total number of trials. As the number of trials grows, experimental probability tends to approach theoretical probability.

For any two events \( A \) and \( B \), the addition rule states: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). The subtraction prevents double-counting outcomes in both events. For mutually exclusive events, \( P(A \cap B) = 0 \), so the rule simplifies to \( P(A \cup B) = P(A) + P(B) \).

Example 14.3.1
Theoretical probability with marbles
  • ProblemA bag contains 5 red, 3 blue, and 2 green marbles. One marble is drawn. Find: (a) \( P(\text{red}) \), (b) \( P(\text{not blue}) \), (c) \( P(\text{red or green}) \).
  • ThinkThere are 10 marbles total. Use the theoretical probability formula for each part.
  • Solve (a) P(red) = 5/10 = 1/2 (b) P(not blue) = 1 - 3/10 = 7/10 (c) Red and green are mutually exclusive: P(red or green) = 5/10 + 2/10 = 7/10
  • CheckAll probabilities fall between 0 and 1. ✓
Answer: (a) 1/2, (b) 7/10, (c) 7/10.
Section 14.4

Independent and Dependent Events

Two events are independent if the occurrence of one does not change the probability of the other. Flipping a coin twice is a classic example — the first flip has no effect on the second. Two events are dependent if the occurrence of one changes the probability of the other. Drawing cards without replacement is a classic example: removing one card changes the deck composition.

Multiplication Rules

Independent events: \( P(A \cap B) = P(A) \cdot P(B) \)

Dependent events (general rule): \( P(A \cap B) = P(A) \cdot P(B \mid A) \)

For independent events, \( P(B \mid A) = P(B) \), so the general rule recovers the simpler form. For three or more independent events: \( P(A \cap B \cap C) = P(A) \cdot P(B) \cdot P(C) \).

Example 14.4.1
Dependent events — drawing without replacement
  • ProblemA box contains 4 red chips and 6 blue chips. Two chips are drawn without replacement. Find the probability that both are red.
  • ThinkThe first draw changes what remains. These events are dependent. Use the general multiplication rule.
  • Solve P(first red) = 4/10 P(second red | first red) = 3/9 = 1/3 (Only 3 red chips remain among 9 total.) P(both red) = (4/10) · (3/9) = 12/90 = 2/15
  • Check\( 2/15 \approx 0.133 \). Drawing two reds from a mostly blue box is unlikely but possible. ✓
Answer: The probability that both chips are red is \( \frac{2}{15} \approx 13.3\% \).
Section 14.5

Conditional Probability

Conditional probability is the probability of an event given that something else is already known to be true. The notation \( P(B \mid A) \) is read "the probability of \( B \) given \( A \)." Knowing that \( A \) occurred restricts the sample space — we now consider only outcomes where \( A \) happened and ask what fraction also include \( B \).

Conditional Probability Formula

\[ P(B \mid A) = \frac{P(A \cap B)}{P(A)}, \quad P(A) \neq 0 \]

Testing independence: Two events are independent if and only if \( P(B \mid A) = P(B) \). If this equality holds, knowing \( A \) gives no information about \( B \).

Example 14.5.1
Conditional probability from a two-way table
  • ProblemA university surveyed 200 students. The table shows major and internship completion. Find: (a) \( P(\text{internship}) \), (b) \( P(\text{internship} \mid \text{STEM}) \), (c) are these events independent?
  • Data STEM: 60 with internship, 40 without (100 total) Non-STEM: 30 with internship, 70 without (100 total) Total: 90 with internship, 110 without (200 total)
  • Solve (a) P(internship) = 90/200 = 0.45 (b) P(internship | STEM) = 60/100 = 0.60 (c) Since 0.60 ≠ 0.45, the events are NOT independent. STEM majors were more likely to have completed an internship.
Answer: P(internship) = 0.45; P(internship | STEM) = 0.60. These differ, so major and internship status are dependent events.
Student Warning — Reversing Conditional Probability

\( P(B \mid A) \) and \( P(A \mid B) \) are generally different values. Confusing them — called the base rate fallacy — leads to serious errors in medical testing, legal reasoning, and scientific interpretation. Always identify which event is given and which is being predicted.

Practice 14.1 – 14.5Probability Concept Check
  1. A number is chosen at random from \( \{1, 2, 3, \ldots, 10\} \). Find: (a) \( P(\text{even}) \), (b) \( P(\text{greater than 7}) \), (c) \( P(\text{even or greater than 7}) \), (d) \( P(\text{even and greater than 7}) \).
  2. A bag contains 6 white balls and 4 black balls. Two balls are drawn without replacement. Find: (a) \( P(\text{both white}) \), (b) \( P(\text{first white, second black}) \), (c) \( P(\text{at least one black}) \).
  3. A survey of 150 students found that 90 own a laptop, 60 own a tablet, and 30 own both. Find: (a) \( P(\text{laptop}) \), (b) \( P(\text{tablet} \mid \text{laptop}) \), (c) are owning a laptop and tablet independent? Show your work.
Sections 14.6 – 14.8
Counting and Binomial Experiments
Permutations · Combinations · Binomial Probability Formula
Section 14.6

Permutations

A permutation is an arrangement of objects in which order matters. If you are selecting first, second, and third place finishers in a race, different arrangements produce different outcomes.

Before defining permutations formally, recall the Fundamental Counting Principle: if one decision can be made in \( m \) ways and an independent second decision in \( n \) ways, together they can be made in \( m \cdot n \) ways. This extends to any number of independent decisions.

The factorial of a whole number \( n \), written \( n! \), is the product of all positive integers from \( n \) down to 1: \( n! = n \cdot (n-1) \cdots 2 \cdot 1 \). By convention, \( 0! = 1 \).

Permutation Formula

The number of ordered arrangements of \( r \) objects chosen from \( n \) distinct objects:

\[ {_n}P_r = \frac{n!}{(n-r)!} \]

Evaluate by canceling: nPr = n · (n−1) · ... multiply r factors starting from n.

Example 14.6.1
Permutations — filling ranked positions
  • ProblemA club of 10 members needs to elect a President, Vice President, and Treasurer (three different people, positions matter). How many ways can these roles be filled?
  • ThinkOrder matters — different assignments to the same positions are different outcomes. Use the permutation formula with \( n = 10 \), \( r = 3 \).
  • Solve _{10}P_3 = 10! / (10 - 3)! = 10! / 7! = 10 · 9 · 8 = 720
  • Check10 choices for President, 9 remaining for VP, 8 remaining for Treasurer: \( 10 \cdot 9 \cdot 8 = 720 \). ✓
Answer: There are 720 ways to fill the three positions.
Tip — Evaluating Permutations Efficiently

You do not need to compute the full factorial. \( \frac{10!}{7!} \) simply means multiplying \( 10 \cdot 9 \cdot 8 \) — start at \( n \) and take exactly \( r \) factors. Cancel the matching factorial before multiplying.

Section 14.7

Combinations

A combination is a selection of objects where order does not matter. If you are choosing 3 people to serve on a committee from 10, a different arrangement of the same 3 people is still the same committee — choosing Alice, Bob, and Carla is the same as choosing Carla, Alice, and Bob.

Combination Formula

The number of unordered selections of \( r \) objects from \( n \) distinct objects:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Read \( \binom{n}{r} \) as "n choose r." Dividing by \( r! \) removes the effect of order: there are \( r! \) arrangements of any selection, so permutations ÷ \( r! \) = combinations.

Question TypeUseWhy
Filling ranked positions, arranging in sequencePermutationDifferent orderings produce different outcomes
Forming a group, selecting a committee, choosing itemsCombinationAll arrangements of the same group count as one outcome
Example 14.7.1
Combinations — selecting a group
  • ProblemA teacher wants to choose 4 students from a class of 20 to attend a mathematics conference. How many different groups of 4 can be selected?
  • ThinkNo internal ranking exists within the group. Order of selection does not matter. Use a combination with \( n = 20 \), \( r = 4 \).
  • Solve C(20,4) = 20! / (4! · 16!) = (20 · 19 · 18 · 17) / (4 · 3 · 2 · 1) = 116280 / 24 = 4845
  • Check\( 20 \cdot 19 \cdot 18 \cdot 17 = 116{,}280 \) and \( 4! = 24 \). Then \( 116{,}280 \div 24 = 4{,}845 \). ✓
Answer: There are 4,845 different groups of 4 students that could be selected.
Section 14.8

Binomial Situations

A binomial experiment has exactly four characteristics: (1) a fixed number of trials \( n \); (2) each trial has exactly two possible outcomes — success or failure; (3) the probability of success \( p \) is the same on every trial; and (4) the trials are independent. The word "success" names the outcome being counted, regardless of whether it is desirable.

Binomial Probability Formula

The probability of exactly \( k \) successes in \( n \) independent trials, each with probability \( p \) of success:

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

C(n,k) counts which k trials are successes.

p^k gives the probability of k successes.

(1-p)^(n-k) gives the probability of n-k failures.

Example 14.8.1
Binomial probability — free throws
  • ProblemA basketball player makes 70% of her free throws. She attempts 5. Find the probability she makes exactly 3.
  • ThinkFixed trials (\( n = 5 \)), two outcomes (make/miss), constant probability (\( p = 0.7 \)), independent trials. This is binomial with \( k = 3 \).
  • Solve P(X = 3) = C(5,3) · (0.7)³ · (0.3)² C(5,3) = 10 (0.7)³ = 0.343 (0.3)² = 0.09 P(X = 3) = 10 · 0.343 · 0.09 = 0.3087
  • CheckAbout 30.9% — reasonable for a 70% shooter. ✓
Answer: The probability of making exactly 3 out of 5 free throws is approximately 0.309, or 30.9%.
MGU Library Connection
  • The binomial probability formula is foundational in MGU Statistics, where it supports confidence intervals and hypothesis testing
  • Combinations connect to the Binomial Theorem in MGU Precalculus, which expands expressions of the form \( (a + b)^n \)
  • Permutations and combinations underlie sampling theory and the mathematics of randomized experiments
Practice 14.6 – 14.8Counting and Binomial Practice
  1. Evaluate without a calculator: (a) \( 6! \), (b) \( {_8}P_3 \), (c) \( \binom{7}{2} \), (d) \( \binom{10}{10} \).
  2. A four-digit PIN uses digits 0–9 with no repetition. (a) How many PINs are possible? (b) How many PINs begin with 5?
  3. A committee of 5 is selected from 12 candidates. (a) How many committees are possible? (b) How many include both Alice and Bob? (c) How many include Alice but not Bob?
  4. A biased coin lands Heads with probability 0.6. It is flipped 4 times. Find: (a) \( P(X = 0) \), (b) \( P(X = 4) \), (c) \( P(X = 2) \).
  5. A factory has a 5% defect rate. An inspector selects 10 items. (a) What is the probability none are defective? (b) What is the probability exactly 2 are defective?
Sections 14.9 – 14.13
Data, Distributions, and Decision-Making
Distributions · Normal Curve · Regression · Model Comparison · Decision Framework
Section 14.9

Data Distributions

When we collect numerical data, we want to understand the overall pattern: where is the data centered, how spread out is it, and what shape does it take? These questions are addressed by studying the distribution — the pattern of how values are spread across their range.

Shape describes the visual pattern: symmetric (left and right sides mirror each other), skewed right (long right tail, data clusters left), skewed left (long left tail, data clusters right), uniform (values spread evenly), or bimodal (two clear peaks).

Center describes a typical value. The mean is the arithmetic average \( \bar{x} = \sum x_i / n \). The median is the middle value when data is ordered. The mode is the most frequently occurring value.

Spread describes variability. The range is maximum minus minimum. The standard deviation measures how far, on average, values deviate from the mean.

Standard Deviation Formula

\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \]

A larger standard deviation means more variability. A standard deviation of 10 points on an exam means most students scored within about 10 points of the average.

Example 14.9.1
Describing a distribution
  • ProblemTen students scored: 72, 75, 78, 80, 80, 82, 84, 86, 90, 93. Describe the distribution using center and spread.
  • Solve Mean: (72+75+78+80+80+82+84+86+90+93)/10 = 820/10 = 82 Median: average of 5th and 6th values = (80+82)/2 = 81 Mode: 80 (appears twice) Range: 93 − 72 = 21
  • CheckMean ≈ Median ≈ 81–82: the distribution is approximately symmetric. ✓
Answer: Mean = 82, Median = 81, Mode = 80, Range = 21. Approximately symmetric with mild spread.
Section 14.10

Normal Distribution as an Introduction

Many naturally occurring data sets — heights of adults, test scores across large populations, measurement errors — follow the normal distribution. When graphed, the shape resembles a symmetric bell: values cluster around the center, and extreme values become increasingly rare in both directions. The normal distribution is completely described by its mean \( \mu \) and standard deviation \( \sigma \).

The Empirical Rule (68-95-99.7 Rule)

For normally distributed data:

≈ 68% of values fall within 1 standard deviation of the mean: (μ − σ, μ + σ)

≈ 95% of values fall within 2 standard deviations: (μ − 2σ, μ + 2σ)

≈ 99.7% of values fall within 3 standard deviations: (μ − 3σ, μ + 3σ)

Z-score: \( z = \dfrac{x - \mu}{\sigma} \) — the number of standard deviations a value is from the mean.

Diagram — The Bell Curve

Draw a bell-shaped curve centered at \( \mu \). Mark \( \mu - \sigma \), \( \mu \), and \( \mu + \sigma \) on the horizontal axis. Shade the region within one standard deviation and label it 68%. Extend to two standard deviations and label the total shaded region 95%. Extend to three standard deviations and label it 99.7%.

Example 14.10.1
Applying the empirical rule and z-scores
  • ProblemScores on a standardized test are normally distributed with mean 500 and standard deviation 100. (a) What range covers approximately 68% of test takers? (b) What percentage scored above 700? (c) A student scored 350. What is their z-score?
  • Solve (a) 1 standard deviation: (500 − 100, 500 + 100) = (400, 600) ≈ 68% of test takers scored between 400 and 600. (b) 700 is 2 standard deviations above the mean. 95% fall between 300 and 700, so 5% fall outside — 2.5% above 700. (c) z = (350 − 500) / 100 = −1.5 The student's score is 1.5 standard deviations below the mean.
Answer: (a) 400 to 600; (b) ≈ 2.5%; (c) z = −1.5.
MGU Library Connection
  • The normal distribution is central to MGU Statistics: z-scores and areas under the normal curve support confidence intervals and z-tests
  • In MGU Biology, Physics, and Engineering, the normal distribution models measurement error, population variation, and material properties
  • MGU Precalculus extends the use of z-scores into formal probability calculations
Section 14.11

Regression Models

Linear regression finds the line that best fits a data set by minimizing the total squared vertical distance between the line and the data points. This least-squares regression line has the form \( \hat{y} = a + bx \), where \( b \) is the slope (predicted change in \( y \) per unit increase in \( x \)) and \( a \) is the y-intercept (predicted value of \( y \) when \( x = 0 \)).

The correlation coefficient \( r \) measures the strength and direction of the linear relationship. \( r \) always falls between \( -1 \) and \( 1 \): values near \( \pm 1 \) indicate strong linear association; values near 0 indicate weak or absent linear association. The coefficient of determination \( r^2 \) is the proportion of variability in \( y \) explained by the model.

A residual is the difference between an actual data value and the predicted value: \( \text{residual} = y_{\text{actual}} - \hat{y}_{\text{predicted}} \). A residual plot graphs residuals against \( x \)-values. Residuals scattered randomly around zero indicate a good fit; a curved pattern in the residual plot means the chosen model is inappropriate for the data.

Example 14.11.1
Interpreting a linear regression
  • ProblemA regression line for hours studied (\( x \)) vs. exam score (\( y \)) is \( \hat{y} = 55 + 4.8x \) with \( r = 0.91 \). (a) Interpret slope and intercept. (b) Predict score for 6 hours studied. (c) Interpret \( r \).
  • Solve (a) Slope: each additional hour studied predicts +4.8 points. Intercept: a student studying 0 hours is predicted to score 55. (b) ŷ = 55 + 4.8(6) = 55 + 28.8 = 83.8 points (c) r = 0.91 is close to 1 — a strong positive linear relationship. More hours studied is strongly associated with higher scores.
Answer: Slope = 4.8 pts/hr; intercept = 55 pts; predicted score for 6 hrs = 83.8; strong positive linear association (r = 0.91).
Correlation Does Not Imply Causation

A strong correlation coefficient does not prove that one variable causes the other. Both may be influenced by a third variable (a confounding variable), or the association could be coincidental. Regression describes association — not cause and effect. To draw causal conclusions, a carefully designed controlled experiment with random assignment is required.

Section 14.12

Comparing Linear, Quadratic, Exponential, and Logarithmic Models

One of the central skills of mathematical modeling is choosing the right function family to describe a data set. In Algebra II, you have studied four major families. Each describes a different kind of relationship.

ModelFormBehaviorSignal in Data
Linear\( f(x) = a + bx \)Constant rate of changeFirst differences approximately constant
Quadratic\( f(x) = ax^2 + bx + c \)Parabola; rises then falls (or vice versa)Second differences approximately constant
Exponential\( f(x) = ab^x \)Constant percent changeRatios of successive \( y \)-values approximately constant
Logarithmic\( f(x) = a + b\ln(x) \)Grows quickly then levels offRapid initial growth that decelerates

How to Choose a Model

1. Plot the data first. Does it look like a line, a curve that levels off, a curve that accelerates, or a peak-and-fall pattern? 2. Check differences and ratios. Constant first differences → linear; constant second differences → quadratic; constant ratios → exponential; rapid growth that decelerates → logarithmic. 3. Fit candidate models and compare \( r^2 \). Higher \( r^2 \) indicates better fit. 4. Examine the residual plot. A good model produces no systematic pattern. 5. Consider context. Bacterial growth calls for an exponential model before you even look at data. Projectile height calls for quadratic based on physics.

Example 14.12.1
Identifying an exponential model from ratios
  • ProblemApp users (in thousands): Year 1 → 3, Year 2 → 9, Year 3 → 27, Year 4 → 81, Year 5 → 243, Year 6 → 729. Determine the best model.
  • ThinkCompute ratios of successive values to check for exponential behavior.
  • Solve 9/3 = 3, 27/9 = 3, 81/27 = 3, 243/81 = 3, 729/243 = 3 Each term is exactly 3× the previous — constant ratio. Exponential model: f(x) = 3^x Check: f(1) = 3 ✓ f(2) = 9 ✓ f(6) = 729 ✓
Answer: Exponential model \( f(x) = 3^x \). The number of users triples each year.
Example 14.12.2
Using residuals to compare models
  • ProblemTwo models for monthly profit data: Linear \( \hat{y} = 2.1x + 1.0 \) with \( r^2 = 0.82 \) and a curved residual plot. Exponential \( \hat{y} = 1.5 \cdot 1.3^x \) with \( r^2 = 0.97 \) and a random residual plot. Which model is better?
  • ThinkUse both \( r^2 \) and residual plot behavior.
  • Solve r² comparison: 0.97 vs. 0.82 — exponential explains 15% more variability. Residual plot: linear shows curved pattern (systematic error). Exponential shows random scatter (good fit).
Answer: The exponential model is more appropriate. Higher r² and a pattern-free residual plot both confirm better fit.
Section 14.13

Making Decisions from Data

Statistics is not just about computing numbers — it is about using those numbers to reason clearly and make better decisions. This section brings together the ideas of the chapter into a coherent decision-making process.

A Five-Step Decision Framework

Step 1 — Define the question. What do you actually want to know? Be specific.

Step 2 — Collect and examine the data. How was it collected? What shape, center, and spread does it have?

Step 3 — Select and fit a model. What function family best describes the relationship? Check with r² and residual plots.

Step 4 — Interpret results in context. What do the slope, intercept, or growth rate actually mean? Could results be due to chance?

Step 5 — Communicate the conclusion. State what the data supports, what remains uncertain, and what assumptions were made.

Example 14.13.1
Decision from data — comparing tutoring programs
  • ProblemA school compares two tutoring programs. Monthly test averages: Program A (months 1–4): 61, 64, 63, 65. Program B (months 5–8): 71, 74, 76, 79. Which program appears more effective?
  • ThinkCompare means and trends. Also consider study design limitations.
  • Solve Mean (Program A): (61+64+63+65)/4 = 253/4 = 63.25 Mean (Program B): (71+74+76+79)/4 = 300/4 = 75.00 Difference: 75.00 − 63.25 = 11.75 points Program B also shows an increasing trend; Program A is nearly flat.
  • CheckLimitation: programs were used sequentially, not simultaneously. Improvement may partly reflect students becoming more comfortable over time, not just the program itself. A controlled comparison with random assignment would be stronger evidence.
Answer: Program B is associated with higher average scores (+11.75 points) and a positive trend. However, sequential design limits confidence. A controlled experiment would provide stronger evidence.
Practice 14.9 – 14.13Data and Regression Practice
  1. State the mean, median, and mode for: (a) 5, 7, 8, 8, 10, 12, 14; (b) 3, 3, 3, 7, 9, 11, 11.
  2. Scores are normally distributed with \( \mu = 74 \) and \( \sigma = 8 \). (a) What range includes ≈ 68% of students? (b) A student scored 90 — what is their z-score? Is this unusually high? (c) Approximately what percentage scored below 58?
  3. For the data 4, 8, 16, 32, 64 at \( x = 1, 2, 3, 4, 5 \): compute first differences and ratios to determine whether linear or exponential fits better.
  4. A linear model has \( r^2 = 0.73 \) and a curved residual plot. A quadratic model for the same data has \( r^2 = 0.91 \) and a random residual plot. (a) Which fits better? (b) What does the curved residual plot indicate?
  5. A health department tracks weekly flu cases: 50, 100, 200, 400, 800, 1600, 3200, 6400. (a) Is this linear or exponential? Show with ratios. (b) Write an exponential model. (c) Predict cases in week 10. What are the limitations?
Chapter 14 — Back Matter

Common Mistakes, Summary, and Review

Section 14.14

Common Probability and Statistics Mistakes

Mistake 1 — Confusing "Or" with "And"
P(A or B) = P(A) + P(B) always

Correct: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). The subtraction is necessary unless events are mutually exclusive. Forgetting it double-counts outcomes in both events.

Mistake 2 — Ignoring Dependence
P(both Aces, no replacement) = (4/52) · (4/52)

Correct: After removing one Ace, only 3 remain among 51 cards. The correct calculation is \( \frac{4}{52} \cdot \frac{3}{51} \). Removing one card changes the deck — these events are dependent.

Mistake 3 — Permutation When Combination Is Needed
Using nPr to count committee members (where order does not matter)

Correct: Ask whether order matters. Committees have no internal ranking — use a combination. Races, awards, and ranked positions have order — use a permutation.

Mistake 4 — Reversing Conditional Probability
P(disease | positive test) = P(positive test | disease)

Correct: These are different values — this confusion is the base rate fallacy. The probability of a positive test given the disease (sensitivity) is very different from the probability of having the disease given a positive test (positive predictive value). Always use the conditional probability formula carefully.

Mistake 5 — Confusing Correlation with Causation
r = 0.92, therefore X causes Y.

Correct: A strong correlation only shows that two variables move together — not that one causes the other. Both may be driven by a third confounding variable. Causal conclusions require controlled experiments with random assignment.

Mistake 6 — Using a Linear Model for Curved Data
Fitting ŷ = bx + a to data with a clear curve

Correct: If the scatter plot shows a curve, or the residual plot of a linear model shows a curved pattern, a linear model is inappropriate. Use first differences and ratios to identify the correct model family.

Mistake 7 — Applying the Empirical Rule to Non-Normal Data
Using the 68-95-99.7 rule on a skewed or bimodal distribution

Correct: The empirical rule applies only to normally distributed data. Applying it to skewed or irregular data produces unreliable results. Always check the shape of the distribution first.

Mistake 8 — Interpreting the Y-Intercept Out of Context
Concluding that a company with zero years of operation earns $40,000 profit, based on a regression model

Correct: The y-intercept often falls outside the meaningful range of the data. Extrapolation beyond the data range produces mathematically valid but practically meaningless predictions. Report the intercept's mathematical meaning, not an impossible real-world scenario.

Chapter Summary

Key Terms Review

sample space
The set \( S \) of all possible outcomes of an experiment.
event
Any subset of the sample space — a collection of one or more outcomes of interest.
complement
All outcomes not in event \( E \); written \( E^c \). \( P(E^c) = 1 - P(E) \).
theoretical probability
\( P(E) = \text{favorable outcomes} / \text{total outcomes} \), assuming equally likely outcomes.
experimental probability
Relative frequency based on observed data. Approaches theoretical probability as trials increase.
independent events
Events where \( P(B \mid A) = P(B) \) — knowledge of one gives no information about the other.
dependent events
Events where occurrence of one changes the probability of the other.
conditional probability
\( P(B \mid A) = P(A \cap B) / P(A) \). The probability of \( B \) given that \( A \) has occurred.
permutation
An ordered arrangement of objects. \( {_n}P_r = n!/(n-r)! \). Use when order matters.
combination
An unordered selection of objects. \( \binom{n}{r} = n!/(r!(n-r)!) \). Use when order does not matter.
factorial
\( n! = n \cdot (n-1) \cdots 2 \cdot 1 \). By convention, \( 0! = 1 \).
binomial experiment
Fixed \( n \) trials; two outcomes per trial; constant probability \( p \); independent trials.
binomial probability
\( P(X = k) = \binom{n}{k}p^k(1-p)^{n-k} \). Probability of exactly \( k \) successes.
distribution
The pattern of how data values are spread across their range, described by shape, center, and spread.
standard deviation
A measure of how far, on average, data values deviate from the mean. Larger value = more variability.
normal distribution
A symmetric bell-shaped distribution described by mean \( \mu \) and standard deviation \( \sigma \).
empirical rule
For normal distributions: ≈ 68% within 1σ, ≈ 95% within 2σ, ≈ 99.7% within 3σ of the mean.
z-score
\( z = (x - \mu)/\sigma \). The number of standard deviations a value is from the mean.
regression
A method for fitting a function to data to model the relationship between two variables.
residual
Actual value minus predicted value: \( y - \hat{y} \). Measures the error of the model at each point.
residual plot
Graph of residuals vs. \( x \)-values. Random scatter = good fit; curved pattern = poor fit.
correlation coefficient r
Measures the strength and direction of linear association. Always between −1 and 1.
coefficient of determination r²
The proportion of variability in \( y \) explained by the regression model. Ranges from 0 to 1.

Concept Review Questions

  1. What is the difference between a sample space and an event? Give an example of each for rolling two dice.
  2. Explain what it means for two events to be mutually exclusive. Give one example that is mutually exclusive and one that is not.
  3. How does experimental probability differ from theoretical probability, and what happens to experimental probability as the number of trials increases?
  4. What is the multiplication rule for independent events? Why does it take a different form for dependent events?
  5. What does \( P(B \mid A) \) mean in words? How do you compute it using the conditional probability formula?
  6. How does the conditional probability formula help you test whether two events are independent?
  7. What is the difference between a permutation and a combination? Give a real-world example of each.
  8. What four characteristics define a binomial experiment?
  9. What three features describe a distribution, and what does each one tell you?
  10. What does the empirical rule state, and to what kind of distribution does it apply?
  11. What does a z-score represent?
  12. What does the correlation coefficient \( r \) tell you about a regression model? What does \( r^2 \) add?
  13. What is a residual, and why is a residual plot useful for evaluating model fit?
  14. How would you decide between a linear model and an exponential model for a given data set?
  15. Why does a strong correlation not imply causation? Give an example of a confounding variable.

Skill Practice

Part AProbability
  1. \( P(\text{Ace}) \) from a 52-card deck
  2. \( P(\text{red card}) \) from a 52-card deck
  3. \( P(\text{Ace or red}) \) from a 52-card deck
  4. \( P(\text{not a face card}) \)
  5. Two dice rolled: \( P(\text{sum} = 7) \)
  6. Two dice rolled: \( P(\text{both even}) \)
Part BCounting
  1. \( 7! \)
  2. \( {_{10}P_4} \)
  3. \( \binom{8}{3} \)
  4. \( \binom{6}{0} \)
  5. Relay team of 4 from 8 runners (ordered)
  6. Pizza with 3 toppings chosen from 10
Part CBinomial Probability
  1. \( P(X = 2) \), \( n=5 \), \( p=0.4 \)
  2. \( P(X = 0) \), \( n=4 \), \( p=0.3 \)
  3. \( P(X = 4) \), \( n=4 \), \( p=0.5 \)
  4. \( P(X = 1) \), \( n=6 \), \( p=0.2 \)
Part DNormal Distribution and Z-Scores
  1. A data set is normally distributed with \( \mu = 510 \) and \( \sigma = 90 \). (a) What range covers ≈ 95%? (b) Find the z-score for a value of 600. (c) What percent of values exceed 690?
  2. A manufacturing process produces parts with a mean diameter of 5.00 cm and a standard deviation of 0.05 cm. (a) Between what diameters do ≈ 68% of parts fall? (b) A part measures 4.88 cm — how many standard deviations below the mean is this?

Word Problems

  1. A restaurant menu lists 4 soups, 6 entrées, and 3 desserts. How many different three-course meals (one of each) are possible?
  2. A committee of 5 is selected from 12 candidates. (a) How many committees are possible? (b) How many include both Alice and Bob? (c) How many include Alice but not Bob?
  3. A city considers two infrastructure plans. Plan X costs \$2 million in year 1 and increases by \$200,000 per year. Plan Y costs \$1.5 million in year 1 and increases by 12% per year. (a) Write a linear model for Plan X and an exponential model for Plan Y. (b) In which year will Plan Y first cost more than Plan X? (c) What factors beyond cost should the city consider?
  4. A health department tracks weekly flu cases: 50, 100, 200, 400, 800, 1600, 3200, 6400. (a) Is this linear or exponential? Show with ratios. (b) Write an exponential model. (c) Predict cases in week 10 and discuss limitations of this prediction.
  5. A researcher claims that people who eat breakfast score higher on exams, citing \( r = 0.78 \). (a) Is this a strong correlation? (b) Give an example of a confounding variable. (c) What study design would provide stronger evidence of causation?
  6. A field trial tests a new variety of corn. Rows planted with traditional seed yield (in bushels): 88, 91, 94, 87, 93. Rows planted with new seed yield: 95, 99, 102, 97, 101. Compare the two varieties using mean and range. What limitations apply to this comparison?

Error Analysis

Each problem shows a student's error. Identify it and write the correct solution.

Problem 1
A student writes: "The probability it rains and I forget my umbrella are independent, so P(both) = 0.4 + 0.3 = 0.7."

Identify two errors in this statement. What is the correct approach for independent events?

Problem 2
_{10}P_4 = 10! / 4!

Explain the error and give the correct formula and numerical answer.

Problem 3
r = 0.88, so increasing study time causes higher test scores.

Identify and explain the statistical error. What can r = 0.88 legitimately tell us?

Problem 4
A student applies the 68-95-99.7 rule to a strongly right-skewed data set and concludes that 68% of data falls within one standard deviation of the mean.

What is wrong with this conclusion? What must be true about the data for the empirical rule to apply?

Problem 5
P(disease | positive test) = P(positive test | disease)

Why are these values different? Give a simple numerical example where they produce different results.

Chapter 14 Checkpoint

Chapter 14 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 14
Section 1 — Vocabulary and Concepts
  1. Define sample space and event. Give an example of each for a two-coin flip experiment.
  2. What is the difference between independent and dependent events? Give one real-world example of each.
  3. What four characteristics define a binomial experiment?
Section 2 — Probability Skills
  1. A committee of 3 is selected from 9 people. How many different committees are possible?
  2. A coin is flipped and a standard die is rolled. (a) List the sample space. (b) Find the probability of Heads and a number greater than 4.
  3. A bag has 4 red and 6 blue chips. Two chips are drawn without replacement. Find the probability that both are red.
Section 3 — Binomial Probability
  1. A student guesses randomly on each of 5 multiple-choice questions (4 choices each). Find: (a) the probability of getting exactly 4 correct; (b) the probability of getting all 5 correct.
Section 4 — Normal Distribution
  1. Scores on a state exam are normally distributed with \( \mu = 510 \) and \( \sigma = 90 \). (a) Between what scores do approximately 95% of test takers fall? (b) A student scored 600 — find and interpret their z-score.
Section 5 — Regression and Model Comparison
  1. A linear model \( \hat{y} = 5.2x + 18 \) has \( r^2 = 0.79 \) and a curved residual plot. An exponential model \( \hat{y} = 14 \cdot 1.25^x \) has \( r^2 = 0.96 \) and a random residual plot. (a) Which model is better, and why? (b) Using the better model, predict \( y \) when \( x = 6 \).
Section 6 — Conditional Probability
  1. In a survey of 300 adults, 180 report exercising regularly, 120 report eating a healthy diet, and 80 report doing both. (a) Find \( P(\text{healthy diet} \mid \text{exercises regularly}) \). (b) Are exercising regularly and eating a healthy diet independent? Show your work.
Section 7 — Error Analysis
  1. A student writes \( P(A \text{ or } B) = P(A) + P(B) = 0.3 + 0.5 = 0.8 \) when \( P(A \cap B) = 0.1 \). What is the error, and what is the correct answer?
  2. A student computes \( {_8}P_3 = \frac{8!}{3!} \). Identify the error, give the correct formula, and compute the correct answer.
Section 14.15

Algebra II Readiness for Precalculus

In Chapters 1 through 14 of Algebra II, you have built a comprehensive set of mathematical tools. You extended function knowledge to include polynomial, rational, radical, exponential, logarithmic, and trigonometric functions. You learned to analyze functions through transformations, zeros, asymptotes, and graphs. You solved complex equations and systems, including those involving matrices, polynomial theorems, and logarithms. You studied conic sections and connected algebra to geometric curves. You introduced yourself to trigonometry as a language for both triangles and periodic phenomena. You connected algebra to probability, counting, data analysis, and regression modeling.

These skills are not isolated — they form a coherent body of mathematical knowledge that describes quantity, change, uncertainty, and structure. Mathematics is a language, and Algebra II has given you a much larger vocabulary in that language. The courses ahead will teach you to use that vocabulary with increasing precision, creativity, and depth.

Looking Ahead
Precalculus, Trigonometry, Calculus, Statistics, and Beyond

Precalculus deepens nearly every idea from Algebra II: all function families in greater depth, trigonometric functions fully (identities, inverse trig, Law of Sines and Cosines), vectors and polar coordinates, and limits as an introduction to Calculus.

Trigonometry will revisit the unit circle, extend to all six trig functions, prove major identities, and model periodic phenomena with full precision.

Calculus builds directly on function analysis. The derivative of a function measures instantaneous rate of change — a direct extension of the average rate of change from Chapter 1. The integral accumulates change, connecting to series and data ideas.

MGU Statistics will take every idea from Chapter 14 and develop it rigorously: formal hypothesis testing, confidence intervals, sampling distributions, and regression inference.

Physics, Engineering, Finance, and Computer Science at MGU use every major idea from this book — exponential decay for radioactive materials and investment depreciation, polynomial functions in mechanics, rational functions in electrical engineering, and statistical models in experimental science.

MGU Library Connection
  • Appendix M (Probability and Counting Reference) — permutation, combination, and binomial formulas
  • Appendix N (Statistics and Regression Reference) — correlation, r², residuals, model comparison
  • Appendix J (Sequence and Series Formulas) — connects to binomial and geometric series
  • Appendix P (Precalculus Readiness Check) — assess readiness before enrolling in MGU Precalculus
  • Appendix B (Function Parent Graphs) — review all function families studied in Algebra II
  • Appendix L (Trigonometry Reference Sheet) — preparation for MGU Precalculus and Trigonometry
  • MGU Statistics prerequisite checklist — review before enrolling in MGU Statistics
  • MGU Dictionary entries: probability, conditional probability, normal distribution, regression, binomial, z-score