Double Trouble
Two coins, two dice, a spinner-and-coin combo — when events team up, the sample space multiplies. Students build organized lists, tables, and tree diagrams to count compound outcomes, then design the standard's own random-digit simulation for a question too tangled to count.
Common Core standards covered
- 7.SP.C.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
- 7.SP.C.8.aUnderstand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
- 7.SP.C.8.bRepresent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
- 7.SP.C.8.cDesign and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Teacher-facing overview
Big idea: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation; understand compound probability as the fraction of favorable outcomes in the sample space.
Student target: “I can represent compound sample spaces with lists, tables, and trees, compute probabilities as favorable over total, and design simulations for compound questions beyond direct counting.”
Success criteria: I can list a compound sample space completely, without duplicates. · I can choose the right tool: list, table, or tree. · I can identify the outcomes composing an everyday-language event. · I can design a simulation: digit assignment, trial definition, and enough runs.
Essential question
One die has six outcomes; two dice have thirty-six. How do you catch EVERY outcome of a compound event — and what do you do when the space is too big to draw?
Learning target: I can represent compound sample spaces with lists, tables, and trees, compute probabilities as favorable over total, and design simulations for compound questions beyond direct counting.
Success criteria
- I can list a compound sample space completely, without duplicates.
- I can choose the right tool: list, table, or tree.
- I can identify the outcomes composing an everyday-language event.
- I can design a simulation: digit assignment, trial definition, and enough runs.
Why this matters
7.SP.C.8 completes Grade 7 probability with the multiplication of possibility: compound spaces grow fast, and the representations — lists, tables, trees — are how humans keep count honestly. The HH-HT-TH-TT dispute is the canonical misconception, settled by representation. And C.8c's simulation is a genuinely modern skill: the blood-donor design (the standard's own example) is Monte Carlo thinking, the tool science uses when counting fails.
Students should already be able to
- Simple probability models — Carnival Game Lab.
- Fractions of a whole — favorable over total.
- Organized listing habits.
Vocabulary to define explicitly
- Compound event
An event about the JOINT result of two or more processes.
Example: Rolling doubles — a claim about both dice at once.
Not an example: Rolling a 6 on one die — simple, one process.
🔒 3 more terms with examples and non-examples — Sample space (compound), Tree diagram, Simulation — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- Warm-up tallies and the dispute.
- Odds-sheet spaces during floor work.
- Simulation trial technique.
Summative — the graded evidence: The exit ticket (two compounds and the assignment rationale), scored with the four-criterion rubric.
The task students work on
The games committee needs odds for opening night's compound games. THE DOUBLE FLIP: two coins, win on double heads — and a committee member insists the odds are 1-in-3. THE DOUBLES GAME: two dice, win on any doubles. THE COMBO: spin a three-color spinner (R, B, G), then flip a coin — win on red-and-heads. THE DONOR QUESTION (from the health fair next door): 40% of donors have type A blood — what's the chance the fair needs FOUR OR MORE donors to find its first type A? Too tangled to count — design the random-digit simulation.
Compound odds sheet — sample space shown for every game.
| Game | Tool | Sample space size | P(win) |
|---|---|---|---|
| Double flip (HH wins) | |||
| Doubles game (two dice) | |||
| Combo (red & heads) | |||
| Donor question (≥4 draws to first A) |
Anticipated student strategies
List settles the small dispute, table scales to 36, tree handles mixture — three tools, chosen by the shape of the compound; simulation extends past all three.
- Systematic listing. The student lists first-coin-then-second in fixed order: HH, HT, TH, TT.
The dispute-settler. Fixed ordering is what guarantees completeness — name the habit.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Tallies lean toward mixed ≈ half. The list explains: HH, HT, TH, TT — FOUR outcomes, and 'mixed' is secretly two of them. P(HH) = 1/4. Representation defeats intuition.
Listen for: The penny-heads-nickel-tails WORLD is different from its reverse — two distinct ways to be mixed, each as likely as HH. The coins' looks don't merge the outcomes.
Tool guide posted: LIST for tiny spaces, TABLE for same-process pairs, TREE for mixed processes. And the quiet pattern: 2×2 = 4, 6×6 = 36, 3×2 = 6 — sizes multiply.
- Every cell/path must be equally likely for favorable/total to work.
- Everyday-language events (doubles, 'at least one head') get UNPACKED into their outcomes first — C.8b's demand.
Listen for: The anti-diagonal: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) — six cells, P = 1/6. Unpacking language into cells is the skill.
🔒 The remaining 4 steps — fully scripted with teacher-says lines, discourse questions, look-fors, and interventions — are in the full lesson. See plans →
Misconceptions and how to respond
- “Two coins have three outcomes, so HH is 1/3.” Mixed happens TWO ways — HT and TH are different worlds. Four outcomes, each 1/4; the tally sheet already told you mixed comes up half the time.
- “For P(doubles), list the six doubles — done: 6 outcomes, P = 1.” — the teacher response is in the full lesson.
- “Simulations are just made-up numbers.” — the teacher response is in the full lesson.
- “'At least 4 donors' means exactly the 4th donor is type A.” — the teacher response is in the full lesson.
🔒 Every misconception ships with a tested classroom response. See plans →
Scaffolded practice
The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 1 partially completed, then 4 independent.
- I doSettle the double flip: full sample space and P(double heads).
🔒 5 more prompts across all three release tiers are in the full lesson. See plans →
Worked solutions and scaffolds for every prompt are in the downloadable kit, included with membership. See plans and pricing.
Exit ticket
- Two coins: P(two tails)? Show the space.
🔒 2 more exit-ticket questions are in the full lesson. See plans →
Worked answers, the completed answer tables, and the scoring rubric are in the downloadable kit, included with membership. See plans and pricing.
Printable handouts in the kit
All handouts are in Printable_Handouts.pdf inside the kit download.
Materials and classroom resources
Student experience
Students enter through a low-floor prompt, build or sketch a model, test a strategy, revise their thinking, collaborate, and present a final solution. The task remains accessible while allowing for sophisticated reasoning.
Detailed facilitation notes
- Step 1: Settle the two-coin dispute with the full sample space. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Build the 36-cell two-dice table; find P(doubles). Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Tree a spinner-and-coin compound. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Design and run the blood-donor simulation. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Tables for two dice, trees for mixed events, lists for small cases; extend with the simulation and the at-least trap.
Language support: “I started by…”, “My model shows…”, and “I know my answer is reasonable because…”.
Extension: Require a second solution, alter a condition, or have students write a challenge for another group.
Accessibility: Manipulatives, enlarged visuals, oral response choices, partner reading, and reduced-copy student pages.
Assessment plan
Formative checks: Strategy choice, accuracy, representation, collaboration, and explanation.
End evidence: One compound probability via a complete sample space, and one simulation design with results interpreted.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.