Simulating compound events

Classroom resources

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Students will be able to

• I can run a simulation of two events and record the results in a frequency table.
• I can find a relative frequency and compare it with the probability.
• I can explain why more trials give a better estimate.

Timing

Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 15 min

Before class

Do the Run board yourself first. Students need the two-dice grid (Grids and tree diagrams), complements, and relative frequency and the law of large numbers (Year 7 Many trials).

The one idea: a simulation is many quick trials; its relative frequency estimates the probability and gets closer as trials grow. The common mistake is concluding from a short run (Ella's mistake: no 12 in 20 rolls, so impossible).

The board is one digital tool. Others for a class task: a spreadsheet (=RANDBETWEEN(1,6) in two columns, a sum column, COUNTIF for the frequency table, filled down 1000 rows), an online dice simulator, or a short Python program. Generative AI can write a simulation, but students should check its code and its frequency table against the grid: an AI may describe results it never ran.

Wider: simulate something with no simple sample space, such as how many packets of cereal you buy to collect all 6 cards.

Running the lesson

1. Predict: ask how many pairs make 7, and how many make 12.
2. Run: ask which totals are rare, and why.
3. Investigate: ask how many 12s Ella should expect in 20 rolls.
4. Modify: ask what happens to the gap between f and P as trials grow.
5. Make: ask for the probability before the Run button.

The hard part (on purpose)

Trusting a short run. Ask: how many times would you expect it in this many trials? Then: is what happened surprising?

When students get stuck

Answers to the checks

1. the most common total: Two dice are rolled 100 times and the totals recorded. Which total will probably come up most often?
   7. More pairs make 7 (six of the 36) than any other total; 2 and 12 have just one pair each.
2. 100 trials: Run at least 100 trials. Which total came up most in the table? Is the relative frequency of 7 close to P(7)? Compare with your prediction.
   
   7 usually comes up most, close to 1/6 of the time. In 100 trials it can be a little off: the next step looks at why.
3. Relative frequency: In 200 simulated rolls of two dice, a total of 7 came up 31 times. What is its relative frequency?
   0.155
   31 ÷ 200 = 0.155, close to 1/6 ≈ 0.167.
4. Find the mistake: Ella runs a short simulation. Click the line where her conclusion goes wrong.
   line 3
   Line 3: in 20 rolls, a 1-in-36 event more often than not doesn't happen at all. P(12) = 1/36 from the grid; a short run can't show it is impossible.
5. Two dice in a spreadsheet: In a spreadsheet, A2 and B2 each hold =RANDBETWEEN(1,6), one dice each. Which formula in C2 simulates the total of two dice?
   =A2+B2
   Add the two dice. One random number from 2 to 12 would make every total equally likely, which two dice don't.
6. 1000 trials: Run 1000 trials or more of two dice. Is the relative frequency of 7 within 0.04 of P(7) now?
   
   With 1000 trials the relative frequency is almost always within about 0.03 of 1/6.
7. At least one head: Two coins: the board shows P(at least one head) = 3/4. Say why, using the four outcomes. Then run 1000 trials or more: does the relative frequency agree?
   
   Only TT has no head, so 3 of the 4 outcomes: 3/4. The relative frequency settles near 0.75.
8. A and not A: In a simulation, A happened 380 times in 500 trials. How many times did not A happen?
   120
   500 − 380 = 120. Relative frequencies 0.76 and 0.24 add to 1, just like complementary probabilities.
9. Expected count: Two dice are rolled 600 times. About how many times do you expect a total of 7?
   100
   1/6 × 600 = 100. A simulation will give a number near 100, not exactly.
10. A frequency table: A simulation of two coins recorded these 20 trials: HT, HH, TT, TH, HH, HT, TT, HH, TH, HT, TT, HH, HT, TH, TT, HH, HT, TH, TT, HH. Fill in the frequency table.
    6, 5, 4, 5
    Tally each outcome: 6 + 5 + 4 + 5 = 20 trials. The relative frequency of HH is 6 ÷ 20 = 0.3, near P(HH) = 0.25.
11. Which estimate: Two classes simulate a total of 7 on two dice. Class A: 9 times in 40 trials. Class B: 170 times in 1000. Which estimate is more reliable?
    Class B
    More trials: 0.17 from 1000 trials is far steadier than 0.225 from 40.
12. Close enough: A simulated relative frequency is 0.31 and the probability from the sample space is 1/3. What should you say?
    They agree well enough
    They are close: the difference is chance. Relative frequency varies, and settles near the probability as the trials grow.
13. Review: or: Review: 30 students, 18 play netball, 10 play hockey, 4 play both. What is P(netball or hockey)?
    24/30
    18 + 10 − 4 = 24, so 24/30 = 4/5.
14. your own simulation: Choose your own event on two dice, work out its probability, then run at least 1000 trials.
    
    Your relative frequency should be close to your probability. If it isn't, check your count of outcomes.

Solutions

One possible solution for each task. Other correct programs are fine.

more trials, other events › 1000 trials

Open task: any program that does what the step asks.

more trials, other events › At least one head

Open task: any program that does what the step asks.

more trials, other events › A and not A

Open task: any program that does what the step asks.

more trials, other events › Expected count

Open task: any program that does what the step asks.

your own simulation

Open task: any program that does what the step asks.

Stretch and support

Stretch: Build the two-dice simulation in a spreadsheet with 1000 rows and a frequency table. Does it agree with the board?

Support: A frequency table template for totals 2 to 12 with columns for count, relative frequency and probability.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand the ideas in ONE lesson only, from the lesson "Simulating compound events".

The ideas:
- Setting up and running compound event simulations: I can Students use digital simulation tools to simulate two-event compound scenarios and record results systematically.
- Using simulation results to determine and describe probabilities: I can Students compare simulated relative frequencies to theoretical probabilities for compound events and describe what the results reveal.

What understanding looks like:
- Sets up and runs a digital simulation for a compound event.
- Records results in an organised frequency table.
- Selects an appropriate number of trials for the simulation.
- Calculates relative frequency from simulation results correctly.
- Compares observed relative frequency to theoretical probability.
- Explains discrepancies using the concept of sampling variation.

Mistakes students often make here:
- A student says the simulation is wrong because 7 came up 15 times in 100, not 17.
- A student reports a count as a relative frequency.
- A student expects the sums 2 to 12 to be equally likely.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Simulating compound events".

Exit ticket

A simulation gives a total of 8 on two dice 68 times in 500 trials. Find the relative frequency and compare it with the probability.

68 ÷ 500 = 0.136. P(8) = 5/36 ≈ 0.139: very close.

What each step is for

• the most common total: Committing to the most likely sum of two dice shows whether they think the sums 2 to 12 are equally likely.
• 100 trials: Running 100 trials of two dice and reading the frequency table shows the middle totals come up most, as the grid predicts.
• reading a simulation: Relative frequency is count ÷ trials; a short run can't show a probability is 0 (Ella); simulating two dice means adding two dice, not one random total.
• more trials, other events: More trials bring the relative frequency close to the probability; a different compound event (coins) behaves the same; A and not A always share the trials.
• simulations: Recording results in a frequency table and judging a simulation, mixed; trusting a short run is the planted slip; 'or' is the review.
• your own simulation: Choosing an event, predicting, simulating and describing the result is the whole cycle of a simulation.

Where it leads

This covers AC9M8P03.

Where it leads: statistics from samples (Year 8 Statistics), and independent events and simulations in Years 9 and 10.