Independent Probabilities

Overview

In this lesson (lesson 2 of 6), students learn what it means for events to be independent and how independence affects probability. Building on earlier work with simple probability experiments and using sample spaces, students will compare theoretical probabilities using event notation and frequency language.

Learning intentions

• Students will understand independent events and distinguish them from dependent events.
• Students will generate a simple theoretical sample space for equally likely outcomes.
• Students will calculate theoretical probabilities on a 0–1 scale for independent events.
• Students will explain conclusions by relating probability to outcome frequency from experiments.

Success criteria

• I can define independent events using everyday examples.
• I can decide whether an event is independent or dependent in a given situation.
• I can calculate the probability of independent events by using equally likely outcomes and sample spaces.
• I can justify my answer using clear mathematical language.

Curriculum links

• Probability: record, describe and analyse frequency of outcomes of simple probability experiments, using appropriate language and the 0–1 probability scale.
• Probability: generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities.
• Probability: understand that the probabilities of all possible outcomes sum to 1.
• Mathematics – working mathematically: begin to express arguments formally in probabilistic settings.

Lesson structure (60 minutes)

1. 0–8 min · Why do (hook). Teacher presents two real-life scenarios: (a) flip a coin then flip it again, (b) roll a die and then take a card from a single-use set where the first outcome changes what remains. Students discuss in pairs: which feels “independent” and why, then share one reason.

2. 8–18 min · Introduce independence. Teacher defines: events are independent when one event does not affect the probability of the other (e.g., coin tosses with no change; die rolls where each roll is fresh). Students sort 6 statements into “independent” and “not independent” and must write a one-sentence justification.

3. 18–30 min · Sample space for combined events. Teacher models building a sample space for two outcomes with equally likely cases: Example: Coin then coin (HH, HT, TH, TT). Teacher reinforces that all possibilities are mutually exclusive and equally likely, and that their probabilities add to 1. Students complete a table-based sample space for “die then die” outcomes represented as ordered pairs (e.g., 1 then 3, 6 then 2). They check totals sum to 1.

4. 30–40 min · Calculate independent event probability. Teacher guides calculation using sample spaces: Example: Probability of getting “heads then heads” = 1/2 × 1/2 = 1/4, shown both from the four equally likely outcomes and by reasoning about independence. Students work on two calculations independently, then compare with a partner:

• Coin then coin: P(HT)
• Die then die: P(sum is 7) using outcomes (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) out of 36.

5. 40–52 min · Mini-investigation: compare theory and frequency. Teacher sets up a quick class experiment using digital random tools or physical counters if available (coin simulators, die spinners, or rolling cards). Each group performs 20 trials for: “heads then heads”. Students record frequency, convert to an experimental probability (frequency/20), and state whether it matches the theoretical probability 1/4 closely enough to “agree” within reason (no overclaiming).

6. 52–58 min · Formal reasoning check (teacher-led whole class). Teacher shows two statements and asks students to decide independently:

• “Rolling a die twice is independent.”
• “Choosing a marble, then choosing another without replacement, is independent.” Students justify using the phrase: “The first outcome does/does not change the probability of the second.”

7. 58–60 min · Exit ticket. Students answer:

• (a) Give one example of independent events from everyday life.
• (b) For two coin flips, what is the probability of getting exactly one head? Show your method briefly.

Resources

• Coin-toss and die-roll cards or a simple digital random generator (no links needed; use locally installed tools).
• Group recording sheets with a 4-outcome table for two coins and a 6×6 grid for two dice.
• Counter tokens for experimental frequencies (optional).
• Whiteboard or slides to show worked sample space examples.
• Exit ticket slips.

Assessment

• Teacher checks during “independence sort” for correct reasoning, not just the label.
• During sample space and calculations, teacher listens for clear counting of equally likely outcomes and correct total probability summing to 1.
• Exit ticket to confirm students can both define independence and compute a combined-event probability.

Differentiation

• Support: provide sentence starters for justifications (“Independent because…”, “Not independent because…”). Offer a partially completed sample space for one of the practice questions.
• Support: include an examples-of-ordered-pairs reminder for die outcomes (first roll, second roll).
• Extension: ask students to compare two scenarios and write a short argument about independence and how it would change if replacement were removed.
• EAL/SEN: allow use of diagrams (tree diagrams or tables) and encourage verbal reasoning before writing; provide a glossary of “independent”, “dependent”, “equally likely”, “sample space”.