Probability Models

Introduction

Probability models help you describe uncertain situations with mathematics. Instead of claiming that you know exactly what will happen next, you identify possible outcomes and assign probabilities that show how likely those outcomes are. A useful model makes its assumptions clear, uses probabilities from 0 to 1, and assigns a total probability of 1 across all possible outcomes.

In Grades 9–10, probability models connect Probability, Statistics, sample spaces, random variables, probability distributions, and simulations. You will learn how to build models for simple and compound events, compare theoretical predictions with experimental data, calculate expected values, and decide whether a model is reasonable.

A probability model is not the real world itself. It is a simplified mathematical description of a chance process. For example, a fair-coin model treats heads and tails as equally likely. That model is useful only if the coin toss is close enough to fair for your purpose.

The image above shows how different probabilities for heads and tails produce different Bernoulli coin-toss models. It is a reminder that a model depends on the probabilities you choose, not only on the list of possible outcomes.

What Is a Probability Model?

Outcomes, Sample Spaces, and Events

An outcome is one possible result of a chance experiment. The sample space is the set of all possible outcomes. An event is any collection of outcomes that answers a question you care about.

For one fair six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. The event “roll an even number” is {2, 4, 6}. Because three of the six equally likely outcomes are favorable, the probability of the event is 3/6 = 1/2.

The same sample space can support many events. “Roll more than 4” is {5, 6}, while “roll a prime number” is {2, 3, 5}. A good model therefore separates the possible outcomes from the particular event you are investigating.

The Probability Rules

For the models in this course, every probability must satisfy two basic conditions. First, the probability of any event is between 0 and 1 inclusive. Second, the probabilities of all mutually exclusive outcomes in the full sample space add to 1.

A probability of 0 describes an impossible event in the model. A probability of 1 describes a certain event in the model. Values between 0 and 1 describe different degrees of likelihood.

For equally likely outcomes, you can use:

Probability of an event = number of favorable outcomes ÷ number of possible outcomes

This formula is appropriate only when the outcomes you count really are equally likely.

Models Are Built from Assumptions

Every probability model contains assumptions. A fair-die model assumes that each face has probability 1/6. A model of two independent coin tosses assumes that the first toss does not change the probability distribution of the second toss.

Ask these questions whenever you use a model: What outcomes are included? Which outcomes are assumed equally likely? Are repeated trials independent? Could the physical process, measurement method, or data collection procedure make some outcomes more likely than others?

Building Models from Symmetry

Symmetry often gives a reasonable theoretical model. For a fair coin, the two faces are treated symmetrically, so heads and tails each receive probability 1/2. For a fair six-sided die, each face receives probability 1/6.

A probability distribution for a fair die gives the same probability to every face. If a die were loaded, the sample space could stay the same while the probability distribution changed.

Suppose a game pays 3 points for rolling a 6 and 0 points otherwise. The probability model for the die is still based on six outcomes, but the scoring rule creates a new random variable with only two possible values: 3 and 0. This distinction between physical outcomes and numerical values becomes important when you study expected value.

When Outcomes Are Not Equally Likely

Do not automatically count outcomes if they do not have equal probabilities. Imagine a spinner divided into regions with different areas. Even if four colors appear, each color does not necessarily have probability 1/4. You need information about region size, observed frequencies, or another justified model.

When probabilities are known for individual outcomes, the probability of an event is found by adding the probabilities of the outcomes in that event.

Experimental Probability and Simulation

A theoretical model starts from assumptions about how a process behaves. An experimental model uses observed data. If an event occurs 42 times in 100 trials, its observed relative frequency is 42/100 = 0.42.

As the number of independent trials grows, relative frequencies often become more stable and tend to stay closer to the underlying probability. This does not mean that short runs must look perfectly balanced. Ten fair coin tosses do not have to produce exactly five heads.

A simulation imitates a chance process. You can use physical tools such as coins, dice, cards, or spinners, or digital random-number generators. The simulation must match the important probabilities and structure of the real situation.

For example, if a basketball player makes 70% of free throws, you could use random digits 0–9 and let seven digits represent a made shot and three digits represent a miss. Repeating the simulation many times can estimate probabilities for events such as “make at least four of five shots.”

Comparing Theory with Data

A model is useful when its predictions are close enough to observed results for the intended purpose. Small differences are expected because of random variation. Large or systematic differences may suggest that the assumptions are wrong or that the experiment contains bias.

Suppose a die is rolled 600 times. A fair-die model predicts about 100 occurrences of each face, but not necessarily exactly 100. If one face appears 210 times while the others appear around 75–85 times, that pattern would give you a reason to question the fair-die model.

Compound Events and Representations

A compound event combines two or more stages or conditions. Examples include tossing two coins, drawing two cards, or rolling a die and then spinning a spinner.

You can represent compound sample spaces with organized lists, tables, or tree diagrams. A tree diagram is especially useful when a process happens in stages.

At each branching point, the probabilities of the next possible outcomes should add to 1. To find the probability of one complete path for independent stages, multiply the probabilities along that path. To find the probability of an event containing several mutually exclusive paths, add the path probabilities.

Independent and Dependent Events

Two events are independent when knowing that one happened does not change the probability of the other. Repeated tosses of an ideal fair coin are commonly modeled as independent.

Events are dependent when one outcome changes what can happen next or changes the probabilities. Drawing two cards from a deck without replacement creates dependence because the first card is no longer available for the second draw.

A probability tree can model either case. In an independent process, branch probabilities often stay the same from stage to stage. In a dependent process, later branch probabilities may change.

Probability Distributions and Random Variables

A random variable assigns a numerical value to each outcome of a random process. A discrete probability distribution lists the possible values of a discrete random variable together with their probabilities.

Consider the sum of two fair six-sided dice. The possible sums range from 2 to 12, but they are not equally likely. There is only one ordered outcome that gives a sum of 2, namely 1 and 1. There are six ordered outcomes that give a sum of 7, so 7 is much more likely than 2.

The bars show the probability mass function for the sum of two fair dice. The shape comes from counting how many ordered pairs produce each sum.

A valid discrete probability distribution must assign a probability from 0 to 1 to each possible value, and all listed probabilities must add to 1.

Expected Value

The expected value is a probability-weighted average of the possible numerical values. For a discrete random variable X,

E(X) = sum of each value multiplied by its probability.

Suppose a game gives you 8 points with probability 1/4 and loses 2 points with probability 3/4. The expected value is 8 × 1/4 + (−2) × 3/4 = 0.5 point per play.

An expected value does not have to be a value you can actually obtain in one trial. It describes the long-run average predicted by the model over many repeated trials under the same conditions.

Model Quality: Assumptions, Bias, and Checking

Probability models are only as useful as their assumptions and evidence. You should check whether the model includes the relevant outcomes, assigns plausible probabilities, treats dependence correctly, and matches available data.

A model may fail because a device is biased, a sample is not representative, trials influence one another, or the process changes over time. A simulation can also fail if its random mechanism does not reproduce the probabilities of the real process.

Use this four-step checking routine:

State assumptions: Write down what you are assuming about fairness, independence, and possible outcomes.
Check the probabilities: Confirm that each probability is between 0 and 1 and that the full distribution sums to 1.
Compare predictions with trials: Run enough trials to see whether observed frequencies are reasonably compatible with the model.
Revise when needed: If evidence repeatedly disagrees with the model, change the assumptions or probability assignments.

Extension: Repeated Random Choices and Emerging Patterns

Repeated simple random choices can produce structured distributions. A Galton board sends balls through rows of pegs. At each peg, a ball moves left or right. If those moves are approximately independent and equally likely, the final positions reflect the number of left and right moves in many repeated stages.

With many rows and many balls, the center bins tend to collect more balls than the extreme bins because there are more left-right paths leading to central positions. This is a powerful example of how a probability model connects local random choices with a large-scale pattern.

A related effect appears when you add more independent dice: the distribution of the total becomes more concentrated around its center and develops a smoother mound-like shape. At Grades 9–10, the important idea is not the formal theorem behind this pattern, but the modeling habit of linking repeated random steps to a distribution of possible totals.

Interactive Tasks

Quiz: Test Your Knowledge

Which statement must be true for every probability in a valid model? (It is between 0 and 1 inclusive) (!It is always greater than 1) (!It is always an integer) (!It must equal one half)

What is the sample space of a chance experiment? (The set of all possible outcomes) (!The event that occurs most often) (!The average of all measurements) (!The set of only favorable outcomes)

For a fair six-sided die, what is the probability of rolling an even number? (One half) (!One sixth) (!Two thirds) (!Five sixths)

What does an experimental probability use? (Observed relative frequencies) (!Only geometric symmetry) (!Only exact algebraic formulas) (!A guaranteed future outcome)

Why can the sums of two fair dice not be modeled as equally likely from 2 through 12? (Different sums have different numbers of outcome pairs) (!The dice have different numbers of faces) (!Every sum has probability zero) (!The total probability is greater than one)

What operation finds the probability of one complete path through independent stages of a probability tree? (Multiply the branch probabilities) (!Subtract the branch probabilities) (!Average the branch labels) (!Square every branch probability)

When are two events independent? (When one does not change the probability of the other) (!When they can never happen together) (!When they always have equal probabilities) (!When each event has probability one)

What is expected value? (A probability weighted average of possible numerical values) (!The most likely outcome in every model) (!The largest possible value) (!The number of trials already completed)

What should the probabilities of all outcomes in a complete discrete distribution add to? (One) (!Zero) (!Two) (!Ten)

What is a strong reason to revise a probability model? (Repeated data strongly disagree with its predictions) (!One short experiment is not perfectly balanced) (!The model contains more than two outcomes) (!A probability is written as a fraction)

Memory Game

Outcome	One possible result of a chance experiment
Sample space	The set of all possible outcomes
Event	A selected collection of outcomes
Simulation	An imitation of a chance process
Independence	A relationship in which one event does not change the probability of another
Distribution	A description of how probability is assigned across possible values
Expected value	A probability weighted average of possible numerical values

Drag and Drop

Match the correct terms.	Topic
Sample space	All possible outcomes
Relative frequency	Proportion of trials in which an event occurred
Tree diagram	Branching representation of a multistage process
Independent events	Events whose probabilities do not change each other
Expected value	Long-run probability weighted average

Match every term with the explanation that best describes its role in a probability model.

Crossword Puzzle

Outcome	What is one possible result of a chance experiment called?
Event	What is a selected collection of outcomes called?
Simulation	What method imitates a chance process to estimate results?
Frequency	What word completes the phrase relative _____?
Independent	What describes events when one does not change the probability of the other?
Distribution	What describes how probabilities are assigned across possible values?

LearningApps

Cloze Text

A probability model begins with a

containing all possible outcomes. An event is a

of outcomes that answers a particular question. In a valid model, the probabilities of all complete outcomes add to

. When equally likely outcomes are justified, probability can be found by comparing favorable outcomes with the

number of possible outcomes. Experimental probability is based on observed

. A simulation should reproduce the important chance structure of the

. In a probability tree, independent path probabilities are found by

. The expected value is a probability weighted

of numerical outcomes.

Open-Ended Tasks

Easy

Coin Toss Experiment: Toss a coin 50 times, record the sequence, calculate the relative frequency of heads after every ten tosses, and compare the results with a fair-coin model.
Probability Photo Hunt: Create a one-page image collection of four everyday situations involving uncertainty and write a short probability-model description for each one.
Sample Space Poster: Design a poster that shows the sample space for rolling a die and spinning a two-color spinner, using an organized list, table, or tree.
Probability Explanation Video: Produce a two-minute video that explains the difference between an outcome, a sample space, and an event using your own example.

Standard

Dice Distribution Project: Roll two dice at least 120 times, graph the sums, calculate experimental probabilities, and compare your graph with the theoretical two-dice distribution.
Interview About Chance: Interview three people about a probability claim they encounter in daily life, such as weather forecasts or sports predictions, and analyze what assumptions the claim seems to use.
Design a Simulation: Build a physical or digital simulation for a real situation with unequal probabilities, explain how your random mechanism represents those probabilities, and test it for at least 100 trials.
Model Check Report: Choose a simple game, state its theoretical probability model, collect data from repeated play, and write a short report judging whether the data are reasonably consistent with the model.

Advanced

Expected Value Game Design: Invent a game with at least three numerical payoffs, calculate its expected value, then redesign one probability or payoff so the expected value changes in a way you can justify.
Dependent Events Investigation: Compare drawing two colored objects with replacement and without replacement, build probability trees for both processes, and explain precisely where the branch probabilities differ.
Probability in the Community: Visit a museum, science center, sports venue, transit hub, or other suitable public place and document one system where uncertainty, frequencies, or predictions matter; create a model and discuss its limitations.
Model Critique Documentary: Produce a short documentary or narrated slide video that investigates a public probability claim, identifies its assumptions and evidence, compares it with data where possible, and proposes a better model if needed.

Learning Assessment

Fairness and Evidence Assessment: A die is rolled 120 times and one face appears 29 times; decide whether this alone proves the die is unfair and justify your reasoning using expected variation and the limits of small samples.
Simulation Transfer Assessment: Design a random-digit simulation for a machine that succeeds 65 percent of the time and explain how your digit mapping preserves the intended probability.
Compound Model Assessment: Compare two-stage experiments with and without replacement, construct a correct representation for each, and explain how dependence changes the probability calculations.
Distribution Reasoning Assessment: Explain why a sum of 7 is more likely than a sum of 2 when two fair dice are rolled, using both outcome counting and the shape of the probability distribution.
Expected Value Decision Assessment: Compare two games with different prizes and probabilities by calculating expected values, then explain one reason expected value alone may not settle a real decision.
Model Revision Assessment: Given theoretical predictions and a large set of experimental results that disagree, identify at least two possible faulty assumptions and propose a revised model or new data collection plan.

Evidence of Learning

Strong evidence of learning includes both correct mathematics and the ability to judge whether a probability model fits a situation.

Knowledge: You can define outcomes, sample spaces, events, theoretical probability, experimental probability, independence, distributions, simulations, and expected value.
Representation skills: You can organize compound sample spaces with lists, tables, and tree diagrams and move accurately between these representations.
Calculation skills: You can calculate probabilities for simple and compound events and expected values for discrete random variables.
Modeling skills: You can state assumptions, build a model, run or design a simulation, and compare predictions with observed relative frequencies.
Products: You can present clear graphs, probability trees, reports, posters, simulations, or videos that communicate your reasoning.
Critical reasoning: You can identify bias, dependence, unequal likelihood, and other reasons a model might fail.
Transfer: You can apply probability modeling to unfamiliar contexts such as games, weather, sports, manufacturing, transport, risk, or scientific experiments and explain the limitations of your conclusions.

OERs on the Topic

The English Wikipedia article on Probability provides a broad reference for probability concepts and links to related topics such as Probability theory, Conditional probability, and Probability distribution.

Linked Learning Areas

Probability models connect algebraic reasoning, data analysis, scientific experimentation, computing, and decision-making. The links below summarize the core learning network for this course.

Probability Models

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