Probability Distributions

Overview

In this lesson (7 of 8) you will move from recording and analysing experimental outcomes to representing probabilities as distributions. Students will compare real-world frequency data with probability ideas, then create a simple distribution from a class survey.

Learning intentions

• Students will interpret probability distributions as lists/tables that show probabilities for each outcome on the 0–1 scale.
• Students will record frequencies from data, then convert them into estimated probabilities.
• Students will represent distributions using tables and probability statements.
• Students will explain fairness and equally/unequally likely outcomes using appropriate probability language.

Success criteria

• I can describe what a probability distribution tells us about each possible outcome.
• I can calculate estimated probabilities from a frequency table.
• I can represent a distribution in a clear table (outcome → probability).
• I can check my distribution makes sense by considering totals near 1 (allowing for sampling).

Curriculum links

• Probability — record, describe and analyse frequency of outcomes using appropriate language and the 0–1 probability scale.
• Probability — understand that the probabilities of all possible outcomes sum to 1.
• Probability — generate theoretical sample spaces for equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities (as a comparison point).

Lesson structure (60 minutes)

1. 0–5 min · Do Now (Quick quiz). Teacher shows 5 short questions on the board; students answer silently, face front, arms crossed, upright and silent. Students then check answers briefly at the end of 5 minutes.

• Q1: If an event is certain, what probability scale value is it?
• Q2: If an event can’t happen, what probability value is it?
• Q3: For 20 equally likely outcomes, what is the probability of one specific outcome?
• Q4: A fair coin—probability of heads?
• Q5: What does “estimated probability” mean compared to “theoretical probability”?

2. 5–15 min · I Do (Model probability distributions). Teacher uses a familiar example (e.g., choosing a colour from a bag of counters) and displays a table with outcomes and estimated probabilities. Students watch and copy a worked example, using dyslexia-friendly reading support (printed table template with large font).

• Teacher states: A distribution is a set of all possible outcomes and how likely each one is.
• Teacher builds from frequency: probability ≈ frequency ÷ total.
• Teacher highlights: if all outcomes are included, probabilities should sum to 1 (or be very close for real data).

3. 15–30 min · We Do (Analyse a real-world dataset). Teacher provides a short dataset (printed and projected) such as “Which transport did you use to get to school?” with counts for 5 options. Students work in pairs for brief turn-and-talk prompts, then whole-class to agree.

• Step A (teacher pause): Students identify all possible outcomes and calculate total trials.
• Step B (turn-and-talk 1 min): “Which outcome is most likely in this distribution? How do you know?”
• Step C: Students compute estimated probabilities for two outcomes, then the class computes the full table together.
• Step D (turn-and-talk 1 min): “Do the probabilities add up to 1? Why might they not exactly?”

4. 30–45 min · You Do (Create a class survey distribution). Students run a quick class survey with one question (teacher chooses one with 4–6 options). For example: “Favourite fruit choice today” (apple/banana/grapes/other) or “Best describes you for sport?” (team player/individual/doesn’t matter/not sure).

• Students record answers quickly on a class tally sheet (teacher or one group collects totals; others copy results).
• Students then calculate estimated probabilities and complete their own distribution table: Outcome | Frequency | Probability (rounded to 3 d.p.).
• Success check: Students sum probabilities and compare to 1.

5. 45–55 min · You Do (Represent and justify). Students write 2–3 probability sentences about their distribution, e.g. “Outcome A is the most likely. Outcome B is less likely.” Teacher circulates and prompts using sentence starters.

• Turn-and-talk (2 min): “How would the distribution change if outcomes were equally likely? What would the probabilities be then?”
• Teacher draws attention to theoretical comparison: if outcomes were equally likely, each probability would be 1 ÷ number of outcomes.

6. 55–60 min · Exit ticket (formative assessment). Students answer individually:

• Given a distribution table (3 outcomes), identify the probability of one outcome and state whether probabilities sum to 1 (or how close).
• One-sentence explanation: “What is a probability distribution?”

Resources

• Do Now quiz slips (or projected questions)
• Dataset handout (transport or similar) with clear table format
• Probability distribution table template (large font, dyslexia-friendly spacing)
• Class survey response options (printed slips or on board)
• Calculators permitted for computing probabilities (as per school policy)
• Exit ticket sheet

Assessment

• Quick quiz at start to check retention of probability language and scales (teacher reviews common errors).
• During We Do: observe pair discussions and check one completed probability calculation per pair.
• During You Do: spot-check distribution tables and probability sums; give immediate feedback on rounding and inclusion of all outcomes.
• Exit ticket to confirm understanding of distributions and the “sum to 1” idea.

Differentiation

• Support: provide sentence starters (“The most likely outcome is… because…”), a probability table template, and an example worked table with frequency to probability conversion.
• Dyslexia-friendly reading options: large-print dataset, colour-coded rows for outcomes, reduced text on slides, and read-aloud of the dataset prompt by teacher.
• Support for calculation: offer a “probability = frequency ÷ total” prompt strip and model rounding to 3 d.p.
• Extension for advanced learners: ask students to compare estimated vs theoretical probabilities and quantify difference (e.g., “How far is P(Outcome) from the expected value if equally likely?”).
• EAL: allow oral rehearsal during turn-and-talk; provide a word bank for “equally likely,” “unequally likely,” “estimated,” “possible outcomes,” and “distribution.”