Chance experiments and simulations: a week of ready-to-teach maths

Five lessons for Year 7 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, result 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 at 100%: the random digit table (Days 3 and 5), one per pair, and result cards (Day 5), one set per group.
4. Open the free interactive unit on the board, and check that a spreadsheet opens on the class computers or tablets for Day 4 (the Day 4 worksheet has printed results if none are free).
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

A probability predicts: a fair coin should land heads about half the time. An experiment tests the prediction, one trial after another, and its relative frequency (the count over the number of trials) is what actually happened. The two rarely match exactly, and that is ordinary: short runs swing a long way from the prediction, while long runs land close to it. A simulation uses random digits or a spreadsheet in place of the real coin or die, so thousands of trials take a moment. Explaining a difference means asking whether the gap is ordinary for that many trials, not whether it exists.

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 results

Write a relative frequency as a fraction and a decimal, such as 12/20 = 0.6, with decimals to three places. Write the prediction first, as a count (about 10 heads in 20 flips), and say how far the result is from it. Coins and dice are needed on Day 1, and a spreadsheet on Day 4.

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

Days 1 and 2 run real experiments and follow the relative frequency as the trials add up; Day 3 simulates by hand with random digits; Day 4 scales up with a spreadsheet; and Day 5 explains the differences between predictions and results. It builds on repeated experiments (AC9M6P02) and assigning probabilities (AC9M7P01), and it leads into simulations of compound events in Year 8 (AC9M8P03).

Materials for the week (one trip)

• From the classroom: pencils, rulers, scissors, calculators, this pack printed at 100%.
• A coin and a die for each pair (Day 1).
• A computer or tablet with a spreadsheet for each pair (Day 4), or the printed results on the Day 4 worksheet.
• Cut-out sheet 1, one per pair: the random digit table (Days 3 and 5).
• Cut-out sheet 2, one set per group: 12 result cards (Day 5).

Words for the word wall

trial, outcome, frequency, relative frequency, running relative frequency, prediction, expected number, observed, variation, simulation, random digit, spreadsheet, long run, fair, biased

Dear families

This week in maths, Year 7 is testing probabilities: making predictions, running chance experiments and computer simulations, and explaining why results and predictions rarely match exactly.

Try this at home

• Flip a coin 20 times together. Predict first, then see how far the result lands from 10 heads.
• Talk about streaks: after five heads in a row, is tails more likely next? (No: the coin has no memory.)
• Count the sixes in 30 rolls of a die during a board game, and compare with the prediction of 5.
• Ask: why would a coin that landed heads 7 times in 10 flips not worry a scientist, when 700 times in 1000 flips would?

Predictions this week

Fill one row a day with an experiment, the prediction, and what actually happened.

Printed from the free seegongsik Chance experiments and simulations teaching pack · seegongsik.com/au/y7/probability/AC9M7P02/pack

Running an experiment

An experiment repeats one chance action as trials. Predict first, then record each result and find the relative frequency: the count over the number of trials. Students flip a coin and roll a die, pool the class’s flips, and compare everything with the predictions.

We are learning to

• carry out repeated trials and record them in a tally,
• find a relative frequency as a count over the number of trials,
• compare observed results with a prediction.

Success criteria

• I can record trials and find a relative frequency as a fraction and a decimal.
• I can say how far a result is from its prediction.

You need

A coin and a die for each pair, the worksheet, and a class table on the board with a row for each pair.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Dividing by the wrong total: 12 heads in 20 flips is 12/20 = 0.6.
• Taking any gap from the prediction as a sign that the coin or die is unfair.
• Changing how the coin is flipped part-way through. Every trial should be done the same way.

Answers

• Part A: answers vary. Predict 10 heads; about 19 pairs in 20 get between 6 and 14.
• Part B: answers vary. Predict 5 of each face; a face usually lands between 2 and 9 times.
• Part C: relative frequencies 0.7, 0.45, 0.35, 0.4, 0.6, 0.55; furthest from 0.5 is pair 1, 0.2 away; the class has 61 heads in 120 flips, 0.508, closer than every pair because more trials land closer in proportion.
• Part D: not yet. A fair coin lands 15 or more heads, or 5 or fewer, in 20 flips about 4 times in 100 runs. Flip it a few hundred more times before deciding.

Running an experiment

Write each relative frequency as a fraction and a decimal.

A. 20 flips

B. 30 rolls

C. Six pairs

Six pairs each flipped a fair coin 20 times.

1. Which pair landed furthest from 0.5?   ____________
2. What is the relative frequency for all six pairs together?   ____________
3. Is the total closer to 0.5 than the pairs’ results? Why might that be?   ______________________

D. Stretch

A pair got 15 heads in 20 flips and says their coin must be biased. Are they right? What would you do before deciding?

The long run

The running relative frequency is the heads so far over the flips so far. It swings early, when one flip is a big share of the total, and settles near the probability as the flips add up. Students work through and graph a printed run of 40 flips.

We are learning to

• work out a running relative frequency after each trial,
• graph it against the number of trials,
• explain why it swings early and settles later.

Success criteria

• I can work out and plot a running relative frequency.
• I can explain why one flip changes it less as the trials add up.

You need

The worksheet, rulers and calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Dividing by all 40 flips instead of by the flips so far.
• Expecting the graph to stay on 0.5 once it gets there.
• Plotting the count of heads instead of the relative frequency, so the line climbs off the grid.

Answers

• Part A: 1: 1, 1; 2: 1, 0.5; 3: 1, 0.333; 4: 1, 0.25; 5: 1, 0.2; 6: 2, 0.333; 7: 3, 0.429; 8: 3, 0.375; 10: 3, 0.3; 15: 5, 0.333; 20: 9, 0.45; 25: 11, 0.44; 30: 13, 0.433; 35: 16, 0.457; 40: 20, 0.5.
• Part B: a line that starts at 1, swings widely over the first few flips, and flattens out as the flips add up.
• Part C: (a) after flip 5, at 0.2; (b) from flip 19 on; (c) 0.5, 0 from 0.5. Early on each flip is a big share of the total, so one flip moves the line a long way.
• Part D: 21/41 = 0.512, a change of 0.012; from flip 1 to flip 2 the change was 0.5.

The long run

A. A run of 40 flips

One run of a fair coin, H for heads and T for tails. Fill in the heads so far and the running relative frequency, to three places.

B. Graph it

Plot each point from Part A and join the points with straight lines.

C. Read the graph

1. After which flip is the running relative frequency lowest?   ____________
2. From which flip on does it stay within 0.1 of 0.5?   ____________
3. What is it after 40 flips? Why does one flip move it so far at the start?   ______________________

D. Stretch

Flip 41 is heads. What is the new running relative frequency, and how much did it change? Compare that with the change from flip 1 to flip 2.

Simulating with random digits

A simulation stands in for the real event: random digits act as coins, spinners and dice when each outcome gets the right share of the ten digits. Students choose digits for events, run simulations from the random digit table, and pool their rows.

We are learning to

• choose digits that give each outcome its probability,
• run a simulation from a table of random digits,
• compare simulated results with predictions.

Success criteria

• I can design a simulation with random digits and explain why the digits fit.
• I can run it and compare the result with the prediction.

You need

Cut-out sheet 1 (the random digit table), one per pair and kept for Day 5, and the worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Digits that give the wrong share, such as 0 to 5 for heads: six digits of ten.
• Counting the skipped digits (0, 7, 8 and 9) as die rolls.
• Using one digit for a chance of 0.25, which needs two digits at a time.

Answers

• Part A: (1) any five digits, such as 0 to 4 for heads; (2) any three, such as 0, 1 and 2; (3) 1 to 6 for the faces, skipping 0, 7, 8 and 9; (4) one digit gives only tenths, so read two at a time: 00 to 24 is 25 of the 100 pairs.
• Part B (predict 12.5 heads and 7.5 red in 25): heads in rows 1 to 20: 14, 8, 11, 11, 11, 17, 10, 15, 15, 13, 14, 11, 14, 13, 12, 14, 7, 14, 12, 15; red: 10, 4, 9, 9, 10, 8, 9, 10, 7, 7, 7, 8, 11, 5, 6, 7, 6, 10, 9, 7. All 20 rows: 251 heads in 500, 0.502.
• Part C (predict 5 of each): faces 1 to 6 land 6, 7, 3, 7, 1, 6 times; furthest from 5 is face 5, 4 away.
• Part D: the pairs are 95, 96, 99, 59, 11, 56, 24, 68, 92, 85, 51, 92; 2 are 24 or less, against a prediction of 3.

Simulating with random digits

Use the random digit table on cut-out sheet 1. Read along each row from left to right.

A. Choose the digits

1. A fair coin.   Heads: __________   Tails: __________
2. A spinner with P(red) = 0.3.   Red: __________   Not red: __________
3. A fair die.   Faces: __________   What do you do with the other digits? __________
4. An event with probability 0.25. Why will one digit not work? What could you do?   ______________________

B. Your row

C. A die from rows 18 to 20

Read rows 18, 19 and 20. Digits 1 to 6 are faces; skip 0, 7, 8 and 9. Stop after 30 rolls.

D. Stretch

An event has probability 0.25. Read row 17 two digits at a time, which gives 12 pairs; 00 to 24 means the event happens. How many times does it happen, and how many did you predict?

Simulating with a spreadsheet

A spreadsheet runs thousands of trials in a moment: =RANDBETWEEN(1,6) rolls a die in a cell, and =COUNTIF counts the sixes. Students run 10, 100 and 1000 rolls, repeat them, and compare the spread of short and long runs.

We are learning to

• run a simulation with a large number of trials in a spreadsheet,
• find relative frequencies for 10, 100 and 1000 trials,
• compare how far short and long runs land from the probability.

Success criteria

• I can set up and run a spreadsheet simulation.
• I can explain why longer runs land closer to the probability.

You need

A computer or tablet with a spreadsheet for each pair (or the printed results in Part B), the worksheet and calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Typing the formula into every cell instead of filling it down.
• Counting the wrong range, such as A1:A100 for 1000 rolls.
• Thinking the spreadsheet is broken when a recalculation changes the results. New numbers are new trials.

Answers

• Part A: answers vary; predict 1/6 of the rolls as sixes: about 1.7 in 10, 16.7 in 100, 166.7 in 1000.
• Part B: 4 in 10 = 0.4, 0.233 from 1/6; 22 in 100 = 0.22, 0.053 from 1/6; 160 in 1000 = 0.16, 0.007 from 1/6; 1678 in 10 000 = 0.168, 0.001 from 1/6.
• Part C: answers vary. In about 95 runs in 100, the sixes land between 10 and 24 in 100 rolls (0.1 to 0.24), and between 144 and 190 in 1000 rolls (0.144 to 0.19), so the 1000-roll results bunch far more tightly.
• Part D: =IF(RAND()<0.3,1,0) gives 1 with probability 0.3; predict 0.3 × 500 = 150 reds in 500 spins.

Simulating with a spreadsheet

A. Roll a die 1000 times

1. In cell A1, type =RANDBETWEEN(1,6). Fill it down to A1000: that is 1000 rolls.
2. In C1, C2 and C3, count the sixes: =COUNTIF(A1:A10,6), =COUNTIF(A1:A100,6) and =COUNTIF(A1:A1000,6).
3. Record the results, then work out each relative frequency and how far it is from 1/6 (about 0.167).

B. No computers? Use these results

One spreadsheet run of 10 000 rolls, with the sixes counted after 10, 100, 1000 and 10 000 rolls. Fill in the last two columns.

C. Again and again

Make the spreadsheet recalculate five times (type anything in an empty cell and press Enter). Each time, write the relative frequency of sixes in 100 and in 1000 rolls.

D. Stretch

In a new column, =IF(RAND()<0.3,1,0) filled down 500 rows simulates a spinner that lands on red with probability 0.3. Why does the formula work? How many reds do you predict? Run it and compare.

Explaining the differences

A gap between prediction and result is expected; what matters is whether it is ordinary for that many trials.

We are learning to

• decide whether a gap between prediction and result is ordinary for the number of trials,
• explain differences using variation and the number of trials, and what more trials would show.

Success criteria

• I can say whether a result is ordinary variation or worth checking, and why.
• I can explain why a short run can land far from its prediction.

You need

Result cards from cut-out sheet 2 (one set per group, cut out before the lesson), the random digit table, and the quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 1; start the second with problem 2.

Watch for these ideas

• Calling every gap a sign of an unfair device, or never calling any gap one.
• Judging the gap alone: 2 heads off in 10 flips is ordinary, 100 off in 1000 is not.
• Thinking a run of heads makes tails “due” next.

Answers with working

1. ordinary (how often a fair device lands this far or further): A 14%, C 70%, D 12%, G 34%, I 60%, J 25%, L 42%; worth checking: B, E, F, H and K, each under 0.1%.
2. predict 30; 27/90 = 0.3, which a fair spinner matches or passes about 58 times in 100: ordinary.
3. predict 100 each; face 5 is furthest, 14 away, which a fair die does about 14 times in 100: no reason to doubt it.
4. the 2000 trials: 0.247 is 0.003 from 0.25, while 20 trials often stray as far as 0.35; long runs land closer.
5. 5: each flip is still 1/2, and the coin does not remember the 8 heads.
6. not necessarily: 30% of 10 days predicts 3, and 4 is ordinary; judge it over many more days.
7. row 19 has 8 digits that are 0 or 1, so 8 wins in 25 games, against a prediction of 5.
8. 1 ÷ √n = 0.05 gives √n = 20, so about 400 flips.

Explaining the differences

For each problem: find the prediction, compare it with the result, and say whether the gap is ordinary for that many trials.

Problems (7 and 8 are the stretch)

1. Sort the result cards from cut-out sheet 2 and write the letters.

   Ordinary variation	Worth checking

   One reason for your sorting: ______________________________________________
2. A spinner with a 1/3 chance of blue was spun 90 times and landed on blue 27 times. What was the prediction? Is the gap ordinary?
3. A die rolled 600 times gave faces 1 to 6 these counts: 99, 107, 92, 110, 86, 106. Which face is furthest from the prediction? Is the die likely to be fair?
4. An event has probability 1/4. A simulation of 20 trials gave a relative frequency of 0.35, and one of 2000 trials gave 0.247. Which is better evidence about the probability, and why?
5. A coin landed heads 8 times in 10 flips. How many heads do you predict in the next 10 flips? Explain.
6. A forecast gave a 30% chance of rain on each of 10 days, and it rained on 4 of them. Was the forecast wrong?
7. Stretch. A game is won with probability 0.2. Use row 19 of the random digit table, with 0 and 1 meaning a win, to simulate 25 games. How many wins, and how many did you predict?
8. Stretch. On the unit, the gold band for n flips reaches 1 ÷ √n either side of 0.5. About how many flips make the band reach only 0.05 either side?

Random digit table

Days 3 and 5. One copy per pair; keep it after Day 3. Each digit from 0 to 9 is equally likely, and each one is independent of the others. Read along a row from left to right; the gaps are only there to help you keep your place.

Result cards

Day 5. Cut out the 12 cards. Sort them into two groups: ordinary variation, and worth checking.

What we know: Chance experiments and simulations

Work on your own. Show your working, and give decimals to three places.

1. A spinner was spun 40 times and landed on green 14 times. What is the relative frequency of green, as a fraction and a decimal?   ____________
2. A fair coin is flipped 300 times. Predict the number of heads. Would 162 heads surprise you?   ____________
3. A run of flips begins T H H T H. Write the running relative frequency of heads after each flip.   ____________
4. Which digits could you use to simulate an event with probability 0.4?   ____________

5. Simulate 12 trials of the event in question 4, with 0, 1, 2 and 3 meaning it happens, using these digits: 7 2 9 0 4 1 8 3 5 5 6 0. What is the relative frequency?   ____________
6. A spreadsheet simulated 1000 die rolls and counted 181 sixes. What was the prediction, what is the relative frequency, and is the result surprising?   ____________
7. Kim flipped a coin 10 times and got 4 heads. Jo flipped a coin 1000 times and got 470 heads. Whose relative frequency is closer to 1/2? Why would you expect that?
8. A spinner with four equal sectors was spun 80 times and landed on red 14 times, not 20. Explain the difference.

Answers and marking guide

Answers with working

1. 14/40 = 7/20 = 0.35.
2. 150 heads. No: 162 is 12 more, and a fair coin lands this far or further about 18 times in 100.
3. 0, 0.5, 0.667, 0.5, 0.6.
4. Any 4 of the ten digits, such as 0, 1, 2 and 3.
5. 5 of the 12 digits are 0 to 3, so 5/12 = 0.417.
6. About 167 (1/6 of 1000); 181/1000 = 0.181. Not surprising: a fair die lands this far or further about 23 times in 100.
7. Jo’s: 0.47 is 0.03 from 1/2, Kim’s 0.4 is 0.1 away. More trials land closer in proportion.
8. 20 is an average, and results vary: a fair spinner lands this far or further about 15 times in 100. Many more spins would bring the relative frequency nearer 1/4.

A quick three-level guide

Eight questions, four ideas. A student at standard finds relative frequencies, predicts counts, runs a simulation with digits that fit, and explains a gap as ordinary variation that shrinks, in proportion, as the trials grow.

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: experiments, the long run, random digits, spreadsheets, and explaining the differences.