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Course 221 · Life course 43

Reason with Probability and Chance

A chance-model packet with a complete sample space, combined-event calculations, conditional probabilities and a repeated-trial comparison. Six introductory lessons use supplied practice cases.

Before you start

Use paper or a free text document; every core input is supplied in the lessons.

Understand Statistics in Everyday Claims See everything you’ll need

Free download · Original practice data · No external account required. Download the starter pack

What you’ll leave with

A chance-model packet with a complete sample space, combined-event calculations, conditional probabilities and a repeated-trial comparison.

Start course Start without an account. Sign in to save lesson progress across devices.

Your project

Start with the practice files. Build your project one lesson at a time.

Practice files are free without an account. Sign in to save your project record in My Proof: a description, checklist, and optional work link. Keep the original files in your own app or notebook; saving a project record does not upload them.

  1. Get your materialsDownload practice filesSupplied practice cases, a blank working record, complete lesson instructions, and annotated worked examples.
  2. Make your projectView the project checklist Work in your own notebook, document, or software. Use the supplied practice examples when you prefer.
  3. Keep a private recordSave a project record Sign in to save a description and checklist, then return to update them.
Course handbook

Know the route before you begin.

Here’s what you’ll need and what you’ll make. You can return to this guide during any lesson.

Who this is for

Learners aged 14+ building subject foundations and practical independence. Adults can use the same supplied cases.

Prepare before lesson one
  • Use paper or a free text document; every core input is supplied in the lessons.
  • A basic calculator is optional; show the method, units and checks.
  • Suggested earlier course: Understand Statistics in Everyday Claims
Your final project

Finish with proof you can use.

A chance-model packet with a complete sample space, combined-event calculations, conditional probabilities and a repeated-trial comparison.

Save a project record in My Proof
See project checklist and review criteria

Keep your working files in your own notebook or software. Save a private description, checklist, and work link in My Proof.

What to include

  • List a complete equal-outcome sample space.
  • Calculate a complement and an independent combined event.
  • Audit overlap in an inclusive-or event.
  • Compare conditional and unrestricted denominators.
  • Explain observed frequency versus theoretical chance with all six changed cases.

Review and revise

Use these criteria to check your work and choose what to improve.

  • Method: applies the course-specific rules to the supplied cases with visible intermediate work.
  • Accuracy: results agree with the stated facts, units, boundaries and independent checks.
  • Reasoning: explains why the result follows and identifies a limitation or unresolved question.
  • Transfer: completes changed cases and revises the final project when a relevant assumption changes.
Three ideas to help you along the way
Three useful ideas

The rules behind the steps.

01

Observe the actual evidence

A sample space lists all possible outcomes under a stated model. Counts become probabilities only when outcomes are equally likely. Define the experiment and the outcome being counted before dividing.

02

Choose the method that fits the task

Events are independent when learning that one occurred does not change the probability of the other. For independent events, the chance of both is the product. Independent is different from mutually exclusive: independent events can occur together.

03

Retain the limit or revised meaning

A theoretical model describes chance under assumptions; observed relative frequency describes a particular record. Random variation means a short record need not equal the model exactly. More trials do not guarantee exact equality or prove fairness by themselves.

What do these words mean?
Key words

Key terms worth knowing.

Sample space
All outcomes in the stated model.
Complement
Outcomes in which an event does not occur.
Independent
One event does not change another event’s probability.
Disjoint
Events with no shared outcomes.
Conditional probability
Chance within outcomes meeting a stated condition.
Relative frequency
Observed event count divided by the recorded trial count.
Your route

6 lessons.
One useful project.

Work in order the first time. Each lesson makes something the next lesson can use.

  1. 1
    Up next

    Build a complete sample space

    List every outcome for a two-stage experiment and identify when equal-outcome counting applies.

  2. 2
    9 min

    Separate impossible, certain and likely

    Interpret zero, one and intermediate probabilities and calculate a complement.

  3. 3
    10 min

    Combine independent events

    Calculate a combined probability only after checking independence.

  4. 4
    11 min

    Distinguish overlapping from disjoint events

    Calculate an inclusive-or event while removing double-counted overlap.

  5. 5
    11 min

    Update a chance using a condition

    Choose a conditional denominator and distinguish the reverse conditional question.

  6. 6
    13 min

    Compare theoretical chance with repeated trials

    Compare a chance model with observed frequency and explain small-sample variation.

Use the tools

Tools to help you practice.

See all tools
ZIP · practice files

Reason with Probability and Chance practice files

Supplied practice cases, a blank working record, complete lesson instructions, and annotated worked examples.

Text worksheet

Your course project worksheet

A downloadable text worksheet with each lesson’s working output, space for your answers, and the shared final-project checklist. Save your own copy.

Source transparency

Sources and further study.

Explore the references behind the lessons and take your learning further. Source notes explain access and reuse.

OpenStax: probability terminology
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

OpenStax: independent and mutually exclusive events
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

OpenStax: probability rules
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

Ready when you are

Start with the first useful move.

You’ll always know what to do, what to make, and where to go next.

Start lesson one