Simulations and chance experiments: a week of ready-to-teach maths

Five lessons for Year 9 Probability. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus a random digit table, 12 compare cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair: sheet 1 is the random digit table (Days 2 and 3), sheet 2 the compare cards (Day 5). Give each pair a row number of the table to start from.
4. Each pair needs three coins on Day 1 and a computer or tablet with a spreadsheet on Day 4.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

A simulation copies a chance situation with a device whose chances match, such as a random digit or a spreadsheet’s random number, repeats it many times and uses the relative frequency as an estimate of the probability. More trials give a steadier estimate. Comparing a simple event (a head on one toss, 1/2) with a related compound event (at least one head in three tosses, 7/8) shows how combining events changes a chance: “at least one” in several tries is more likely, “every time” is less likely, and some compound events land exactly on the simple chance.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these answers

Theoretical probabilities are fractions in lowest terms found by listing every outcome, as in 7/8. Simulated results are the count over the trials and then a decimal to three places, as in 53/100 = 0.53, with ≈ when the decimal is rounded. Results that come from your own trials vary; the answers give the theory to compare them with.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 runs a chance experiment by hand and pools the class; Day 2 copies a coin and a die with random digits; Day 3 designs a simulation of a new situation in five steps; Day 4 moves the trials onto a spreadsheet, where thousands take seconds; and Day 5 compares simple and compound events and describes results. It builds on listing outcomes and on relative frequency from data earlier in Year 9, and it completes the Year 9 probability strand.

Materials for the week (one trip)

• From the classroom: three coins per pair (Day 1), a computer or tablet with a spreadsheet per pair (Day 4), scissors, this pack printed.
• Cut-out sheet 1, one per pair: the random digit table (Days 2 and 3). Keep it after Day 2.
• Cut-out sheet 2, one per pair: 12 compare cards (Day 5).
• No computers on Day 4? The plan has a version with the digit table and the printed spreadsheet runs. No other equipment to buy.

Words for the word wall

trial, chance experiment, simulation, random, device, model, success, relative frequency, estimate, theoretical probability, law of large numbers, simple event, compound event, at least one, spreadsheet, formula

Dear families

This week in maths, Year 9 is running chance experiments and simulations: copying a chance situation with coins, random digits or a spreadsheet, repeating it many times, and comparing a single event with a combined one.

Try this at home

• Toss a coin 20 times and count the heads. Toss it 20 more. Did the share of heads get closer to a half?
• Ask: is it more likely to get a six on one roll of a die, or at least one six in four rolls? (At least one six in four rolls: a little over a half.)
• Ask: is it more likely to get a head on one toss, or two heads in two tosses? (One toss: a half, against a quarter.)

My chance week

Fill one row a day. Write an experiment you ran, how many trials, how many successes, and the relative frequency.

Repeated chance experiments

A chance experiment repeated many times gives a relative frequency that estimates the probability, and the estimate steadies as the trials grow. Today the class runs two experiments by hand, a simple event (a head on one coin) and a related compound event (at least one head on three coins), and pools the results.

We are learning to

• conduct a repeated chance experiment and record the results in a table,
• find relative frequencies and compare them with theoretical probabilities,
• explain why pooling the class gives a better estimate.

Success criteria

• I can find the relative frequency of a head and of at least one head in three tosses.
• I can say whether a result is close to the theory, and why a bigger pool is closer.

You need

The worksheet; three coins per pair; a class results table on the board (Part C).

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Expecting exactly 15 heads in 30 tosses, and thinking the coin is unfair when it is 12 or 18.
• Counting heads on the three coins instead of counting the tosses with at least one head.
• Believing a coin “owes” a head after a run of tails.

Answers

• Part A: results vary; the theory is 1/2 = 0.5.
• Part B: results vary; the list of eight outcomes has 7 with at least one head (all but TTT), so the theory is 7/8 = 0.875. The compound event happens more often than the simple one.
• Part C: the class totals vary; with more tosses they usually sit closer to the theory than a single pair’s.
• Part D: at least one head is every outcome but TTT (7/8), more than 1/2; three heads is only HHH (1/8), less than 1/2. Combining events can make a chance bigger or smaller.

Repeated chance experiments

A. The simple event: a head on one coin

Toss one coin 30 times. Tally the heads and the tails.

B. The compound event: at least one head on three coins

Toss three coins together 30 times. Tally the tosses with at least one head. List the eight outcomes of three coins to find the theory.

C. Pool the class

Is the class result closer to the theory than your pair’s? Write one sentence.

D. Stretch: bigger or smaller?

Use the eight outcomes to explain why at least one head on three coins is more likely than a head on one coin, but three heads is less likely.

A device that matches

A simulation needs a device whose chances match the real event. The digits 0 to 9 of a random digit table are equally likely, so any five digits make a 1/2 chance and any three a 3/10 chance; for a die, use the digits 1 to 6 and skip the rest. A device with the wrong chances simulates the wrong situation, however many trials it runs.

We are learning to

• choose digits whose chance matches an event, skipping digits when needed,
• simulate a coin and a die with a random digit table,
• spot a simulation design whose device does not match the event.

Success criteria

• I can say which digits mean success, which mean failure and which are skipped.
• I can explain why “a head means a six” does not simulate a die.

You need

The worksheet; the random digit table (cut-out sheet 1) with a different starting row for each pair.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the demo die again.

Watch for these ideas

• Using the digits 1 to 6 for a die but counting 0, 7, 8 and 9 as faces instead of skipping them.
• Thinking the success digits must be next to each other, or must include the digit of the event (6 for a six).
• Every pair starting at the same place in the table, so the class pools the same digits again and again.

Answers

• Part A (one choice each; any digits with the right share will do): coin 0, 2, 4, 6, 8; die 6, skip 0, 7, 8, 9; rain 0, 1, 2; spinner 1, skip 0, 5, 6, 7, 8, 9.
• Part B: results vary; the theory is 1/6 ≈ 0.167, about 3 sixes in 20 rolls.
• Part C: 1 matches (five digits of ten, 1/2); 2 matches (each face is one digit of the six used, 1/6); 3 does not match (a six is two digits of the seven used, 2/7, not 1/6); 4 does not match (a head is 1/2, a six 1/6).
• Part D: for example 1, 2, 3 succeed, 4 to 9 fail, skip 0: 3 digits of the 9 used, 1/3.

A device that matches

A. Match the chance

Choose digits from 0 to 9 that give each event its chance. Skip digits only when you need to.

B. Simulate 20 rolls of a die

Start at your row of the digit table. Digits 1 to 6 are the faces; skip 0, 7, 8, 9. Tally 20 rolls.

C. Good design?

Tick the designs that match the event, and say what is wrong with the others.

D. Stretch: a third

Choose digits that give a 1/3 chance. Which digits succeed, which fail, and which do you skip?

Designing a simulation

A good simulation is designed before it is run: name the event and the model behind it, choose a device whose chances match, define one trial and what counts as a success, repeat many times, then estimate, describe and compare. Today the class designs a simulation for a situation no coin or die fits directly.

We are learning to

• plan a simulation in five steps,
• run the trials with random digits and record the successes,
• compare the estimate with the theoretical probability.

Success criteria

• I can say what one trial is and what counts as a success in the free-throw simulation.
• I can compare my estimate with the theory and explain the difference.

You need

The worksheet; the random digit table from Day 2, with each pair starting at its own row.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the demo trials.

Watch for these ideas

• Using one digit per trial, so only one throw is simulated.
• Counting a success only when both throws go in: that is a different event.
• Using the digits 0 to 7 for a 7 in 10 chance: that is eight digits, 8 in 10.

Answers

• Part A: 1 at least one of two throws goes in; each throw goes in 7 times in 10, whatever the other did. 2 digits 0 to 6 go in, 7 to 9 miss. 3 a trial is two digits; a success is at least one digit from 0 to 6. 4 25 trials from your row. 5 successes over 25, compared with the theory.
• Part B: results vary; the demo on row 2 gave 7/10 = 0.7 over its first 10 trials.
• Part C: of the 100 pairs from 00 to 99, only the 3 × 3 = 9 with two misses fail, so the theory is 91/100 = 0.91. Class results vary and should sit closer to it than one pair’s.
• Part D: for example, digits 1, 2 and 3 for the three animals and skip the rest; a trial reads digits until all three have appeared and records how many boxes that took. The average settles near 5.5 boxes.

Designing a simulation

A. Plan it in five steps

A player makes 7 free throws in 10 and takes two throws. Plan a simulation of P(at least one goes in).

B. Run 25 trials

Start at your row. Write each pair of digits, and tick the trials that are a success.

C. Compare with the theory

D. Stretch: collect all three

Each cereal box holds one of three toy animals, each equally likely. Design a simulation to estimate how many boxes it takes to collect all three. Run five trials.

Digital tools

A spreadsheet’s random number turns a simulation into a few formulas: =RANDBETWEEN(0,1) is a coin and =RANDBETWEEN(1,6) a die. Fill a row of formulas down a thousand rows and every row is one trial; the average of a column of 1s and 0s is the relative frequency. Changing any cell runs the whole simulation again.

We are learning to

• write formulas that simulate coins and dice,
• run thousands of trials and read off a relative frequency,
• describe several runs of one simulation: their middle, their spread and the theory.

Success criteria

• I can write the formulas for three coins and for at least one head.
• I can describe five runs and say why none of them is exactly 7/8.

You need

The worksheet; a computer or tablet with a spreadsheet per pair.

Lesson flow (about 60 minutes)

No computers? Pairs run 20 trials of three coins from the digit table (three digits a trial, an even digit a head), the class pools them, and everyone does Part C. Two shorter sessions instead? Split after Part B.

Watch for these ideas

• Filling down to row 1000 from row 2, which is 999 trials, not a thousand.
• Adding the three coins and calling a total of 1 “at least one head” (it is exactly one).
• Expecting a thousand trials to give 0.875 exactly.

Answers

• Part A: =RANDBETWEEN(0,1); =RANDBETWEEN(1,6); =MAX(A2:C2) (or =IF(SUM(A2:C2)>0,1,0)); =AVERAGE(D2:D1001).
• Part B: results vary; one coin near 0.5 and at least one head near 0.875, with the 10 000-row runs the steadiest.
• Part C: 0.884, 0.862, 0.874, 0.884, 0.898; the mean is ≈ 0.88, the runs go from 0.862 to 0.898, and none is further than 0.023 from the theory 0.875. Each run is its own random sample, so each lands a little off.
• Part D: =IF(AND(A2=6,B2=6),1,0) filled down, then the average; the theory is 1/36 ≈ 0.028, far below a six on one roll (1/6).

Digital tools

A. The formulas

Write the spreadsheet formula for each job.

B. Run it

Fill the formulas down for each number of trials. Record the relative frequencies.

C. Five runs

A spreadsheet ran 1000 trials of three coins five times. The relative frequencies of at least one head were:

D. Stretch: a double six

Write a formula for “both dice show a six” using two dice in A2 and B2. Run it, and compare with the theory and with a six on one roll.

Simple versus compound

A compound event built from a simple one can be more likely, less likely or exactly as likely. “At least one” in several tries beats one try; “every time” loses to it; and some compound events, such as two dice showing the same number, land on the simple chance exactly. A good description of a simulation gives the estimate, the number of trials and the theory it is compared with.

We are learning to

• compare the probability of a simple event with a related compound event,
• describe a simulation’s result: the estimate, the trials and the comparison,
• choose the result to trust when two runs disagree.

Success criteria

• I can sort the twelve cards and give both chances on each.
• I can write a description of a simulation that a reader could check.

You need

The worksheet; the 12 compare cards (cut-out sheet 2) per pair; the unit’s self-check quiz.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with Part B and the quiz.

Watch for these ideas

• Believing a compound event is always more likely (or always less) than its simple event.
• Describing a result without the number of trials, so a reader cannot judge it.
• Trusting the run with the more striking answer instead of the run with more trials.

Answers

• Part A: the card key is with the mini-check answers.
• Part B: for example: “In 500 trials, at least one six in four rolls came up 262 times, 262/500 = 0.524. The theory is 1 − (5/6)⁴ ≈ 0.518, so the estimate is close; it is higher than a six on one roll, ≈ 0.167.”
• Part C: report the 2000-trial run, 1041/2000 ≈ 0.521: it is much closer to the theory ≈ 0.518 than the 20-trial run (13/20 = 0.65), because more trials give a steadier relative frequency.
• Part D: at least one six in four rolls is 1 − (5/6)⁴ ≈ 0.518, just over 1/2; at least one double six in 24 rolls is 1 − (35/36)²⁴ ≈ 0.491, just under 1/2. The first bet wins more often than it loses; the second loses more often.

Simple versus compound

A. Compare cards

For each card, list or count the outcomes to find both chances. Write the letter and the two chances in its pile.

B. Describe a result

A class simulated “at least one six in four rolls” 500 times and got 262 successes. Write a description: the estimate, the number of trials, the theory, and how the compound event compares with a six on one roll.

C. Which run do you report?

The same simulation was run twice: 13 successes in 20 trials, and 1041 in 2000. Which result do you report, and why?

D. Stretch: two old bets

In 1654 the Chevalier de Méré asked Blaise Pascal about two bets: at least one six in four rolls of a die, and at least one double six in 24 rolls of two dice. Which bet wins more often than it loses? Use the complement.

Random digit table

Days 2 and 3. One copy per pair; keep it after Day 2. Each digit from 0 to 9 is equally likely and does not depend on the digits before it. Start at your row and read from left to right, carrying on into the next row; the gaps only help you keep your place.

Compare cards

Day 5, one set per pair: 12 cards. Each card has a simple event and a related compound event. Find both chances, then sort: is the compound event more likely, less likely or just as likely?

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

What we know: Simulations and chance experiments

Work on your own. Give theoretical probabilities as fractions and simulated ones as decimals. Show your list or working.

1. A student uses the digits 0 to 3 for heads and 4 to 9 for tails. Does this simulate a fair coin? Why? ______________________
2. To simulate a die with the digits 1 to 6, which digits are skipped? ______________________
3. A spreadsheet uses =RANDBETWEEN(1,4) for each trial. What does it simulate? ______________________
4. Two coins are tossed. List the outcomes and find P(exactly one head). ______________________
5. Is exactly one head in two tosses more likely, less likely or just as likely as a head on one toss? ______________________
6. A simulation of “at least one six in three rolls” gave 172 successes in 400 trials. What is the relative frequency? ______________________
7. The theory for Question 6 is 1 − (5/6)³. Find it, and say whether the simulation looks right. ______________________
8. Why does a simulation with 10 000 trials give a better estimate than one with 50? ______________________

Answers and marking guide

Answers with working

1. No: heads are 4 digits of 10, 2/5, not 1/2.
2. 0, 7, 8 and 9.
3. A spinner with four equal parts (or a four-sided die): each of 1 to 4 has a chance of 1/4.
4. HH, HT, TH, TT; 2/4 = 1/2.
5. Just as likely: both are 1/2.
6. 172/400 = 0.43.
7. 91/216 ≈ 0.421; the simulation’s 0.43 is close, so it looks right.
8. The relative frequency settles as the trials grow, so a long run lands nearer the true probability than a short one.

A quick three-level guide

Eight questions, four ideas. A student at standard matches a device to its event, finds both chances and compares them.

Card key

Compare cards (Day 5): compound more likely: A (1/2 against 3/4), D (1/6 against 11/36), F (1/4 against 7/16), K (1/2 against 7/8), L (1/2 against 3/4); compound less likely: C (1/2 against 1/4), E (1/6 against 1/36), H (1/2 against 1/4), J (1/4 against 1/16); compound just as likely: B (1/6 against 1/6), G (1/2 against 1/2), I (1/6 against 1/6).

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: chance experiments, matching devices, simulation design, digital tools, and simple against compound.