Simulating conditional probability: a week of ready-to-teach maths

Five lessons for Year 10 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 twelve chance cards to match, twelve which-count cards to sort, 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. Cut the 12 chance cards before Day 1 and the 12 which-count cards before Day 2, and keep each set in an envelope.
4. Book computers or tablets with a spreadsheet for Day 3, one per pair if you can; otherwise run it on the board and pairs record.
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 stands in for a chance situation: a model says what the chances are, a random tool (digits, a die, a spreadsheet) plays it out many times, and we count. All week the model is a morning: rain 40% of the time; if it rains, the bus is late half the time; if it is dry, 20% of the time. To estimate the chance of a late bus given rain, we count only the rainy trials, so the denominator is the rainy count. Small runs swing about; more trials settle near the model value, and a conditional estimate needs extra trials because only some of them meet the condition.

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

A digit rule names its digits: 0 to 3 means rain gives a chance of 4 in 10. Estimates are counts over counts first (5/9), then a decimal to two places (0.56). The printed random digits come from a fixed list, so every class gets the same digits and the answers in this pack match.

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 builds a model and runs it by hand; Day 2 counts inside the condition; Day 3 hands the running to a spreadsheet; Day 4 asks how many trials are enough; and Day 5 puts the whole cycle, design, run, interpret and report, into one task. It builds on the language and tables of conditional probability (AC9M10P01) and on Year 9 experiments with relative frequency, and it prepares the simulation and modelling of senior mathematics.

Materials for the week (one trip)

• From the classroom: calculators, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 chance cards (Day 1).
• Cut-out sheet 2, one per pair: 12 which-count cards (Day 2).
• Computers or tablets with a spreadsheet (Day 3). If you have them: coins and dice (Day 1). No other equipment to buy.

Words for the word wall

simulation, model, trial, random digit, outcome, relative frequency, estimate, conditional probability, given, denominator, model value, variation, RAND, COUNTIFS, report

Dear families

This week in maths, Year 10 is learning to simulate chance situations, by hand and with a spreadsheet, to estimate how likely something is once we know something else, and to judge how far to trust the estimate.

Try this at home

• Ask: how could you use a die to make a 1 in 3 chance? (A 1 or a 2 means yes.)
• Ask: a coin gives 7 heads in 10 tosses. Is it unfair? (Not necessarily: small runs swing about.)
• Ask: to estimate how often the bus is late when it rains, which mornings do you count? (Only the rainy ones.)
• Ask: a forecast says a 40% chance of rain. Over 100 days like this, about how many would be rainy?

My simulation week

Fill one row a day. Write a chance you heard about and how you could simulate it with a coin, a die or random digits.

A model you can run

A simulation needs a model and a fair way to play it out. Students turn chances into digit, coin and die rules, run twenty two-stage mornings by hand from a table of random digits, and match six chances to fair tools.

We are learning to

• turn a chance into a fair rule for digits, coins or a die,
• run a two-stage trial where the second rule depends on the first result,
• record trials so they can be counted later.

Success criteria

• I can say why digits 0 to 3 give rain 40% of the time.
• I can run a morning: the weather digit first, then the bus digit with the rule for that weather.

You need

A coin and a die per pair if you have them; the chance cards (cut-out sheet 1), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Using one bus rule for both: after rain 0 to 4 is late, after a dry morning only 0 or 1.
• Off by one: 0 to 4 is five digits, so 50%, not 40%.
• Reusing a digit: each stage takes a fresh digit.

Answers

• Part A: rain and late 2, 4, 12, 18, 19; rain and on time 1, 3, 6, 7; dry and late 5, 8; every other morning dry and on time.
• Part B: 9 rainy, 5 late: 5/9 ≈ 0.56 against 0.5; 11 dry, 2 late: 2/11 ≈ 0.18 against 0.2. Twenty mornings land near the model, not on it.
• Part C: the card key is with the mini-check answers.
• Part D: pairs 00 to 24 for 25%, and 00 to 34 for 35%.

A model you can run

A. Run twenty mornings

Fill in the weather and the bus for each morning. The first two are done for you.

B. Count

C. Match the tools

Match each chance card to the tool card that makes it fairly. Write the letters.

D. Stretch: two digits at a time

Count inside the condition

A conditional estimate counts only the trials where the condition held. Students complete a table of 200 simulated mornings, estimate the chance of a late bus given rain and of rain given a late bus, compare each with the model, and sort twelve questions by the count that goes on the bottom.

We are learning to

• tally simulated trials in a two-way table,
• divide by the number of trials where the condition held,
• compare an estimate with the model value and explain the gap.

Success criteria

• I can estimate P(late | rainy) from 200 mornings and say why I divided by 81.
• I can say which count goes on the bottom for P(rainy | late).

You need

Calculators; the Day 1 worksheets; the which-count cards (cut-out sheet 2), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Dividing by all 200: that estimates P(rainy and late), not P(late | rainy).
• Expecting the model value exactly: an estimate lands near it, and a little differently each run.
• Using the wrong total: P(rainy | late) divides by the late mornings.

Answers

• Part A: rainy 81, dry 119; late 67, on time 133; 200 in all.
• Part B: P(late | rainy) ≈ 41/81 ≈ 0.51 (model 0.5); P(late | dry) ≈ 26/119 ≈ 0.22 (model 0.2); P(rainy | late) ≈ 41/67 ≈ 0.61 (model 0.2 ÷ 0.32 = 0.625); 41/200 ≈ 0.20 estimates P(rainy and late).
• Part C: the card key is with the mini-check answers.
• Part D: the 200, built on 81 rainy mornings against 9; about 400 of 1 000 mornings would be rainy.

Count inside the condition

A. Two hundred mornings

The same model was run for 200 mornings from a longer list of random digits. Fill in the totals. The rainy row is done for you.

B. Estimate, then compare with the model

C. Sort the cards: what goes on the bottom?

Write each card’s letter under the count you would divide by.

D. Stretch: twenty or two hundred?

Simulate in a spreadsheet

A spreadsheet runs hundreds of trials in a moment. Students build the two-stage model with RAND and IF, count inside the condition with COUNTIF and COUNTIFS, run it three times, and change the model to see what moves.

We are learning to

• build a two-stage simulation with RAND and IF,
• count the trials inside a condition with COUNTIF and COUNTIFS,
• run a simulation again and describe how the estimates vary.

Success criteria

• I can explain what =IF(RAND()<0.4,"rain","dry") does.
• I can estimate P(late | rainy) from 200 spreadsheet mornings.

You need

A computer or tablet with a spreadsheet for each pair, or one on the board with pairs recording; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• A bus formula that ignores the weather: the IF must look at column B first.
• Dividing by 200: the estimate divides by the rainy count, cell E2.
• Expecting the same answer each run: every recalculation is a fresh run.

Answers

• Part A: C2 checks the weather in B2 first, then uses a fresh RAND with the late chance for that weather.
• Part B: each run has about 80 rainy mornings (0.4 × 200) and an estimate near 0.5, often between about 0.4 and 0.6; the runs differ.
• Part C: change 0.2 to 0.3 in C2. P(late | rainy) stays near 0.5; P(rainy | late) falls from 0.625 to about 0.53, as more late mornings are now dry.
• Part D: switching wins whenever the prize is not behind door 1, about 2 runs in 3; sticking wins about 1 in 3.

Simulate in a spreadsheet

A. Build it

Type headings Trial, Weather and Bus in A1, B1 and C1, and 1 to 200 down column A. Then type these, and fill B2 and C2 down to row 201.

B. Run it three times

C. Change the model

D. Stretch: three doors

More trials, steadier estimates

A small run swings about; as trials pile up, an estimate settles near the model value. Students plot a running estimate of P(late | rainy) from 500 rainy mornings, compare groups of 10 with groups of 50, and work out how many mornings a conditional estimate needs.

We are learning to

• plot a running estimate and describe how it settles,
• compare the spread of estimates from small and large runs,
• explain why a conditional estimate needs more trials than it seems.

Success criteria

• I can describe how the estimate moves from 10 to 500 rainy trials.
• I can work out how many mornings give 500 rainy ones.

You need

Calculators; rulers; the worksheet; the Day 3 spreadsheet if pairs can reach it.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Judging a model on a few trials: 10 trials can land far from it.
• Expecting the estimate to reach 0.5 exactly: it closes in and keeps wobbling.
• Counting every morning: only the rainy mornings feed this estimate.

Answers

• Part A: 2/10 = 0.2, 8/20 = 0.4, 26/50 = 0.52, 54/100 = 0.54, 102/200 = 0.51, 254/500 = 0.508. The line swings early, then settles close to 0.5.
• Part B: groups of 10 from 0.2 to 0.7, a range of 0.5; groups of 50 from 0.38 to 0.58, a range of 0.2. Bigger groups vary less.
• Part C: 400 of 1 000; 500 ÷ 0.4 = 1 250 mornings; at 10% rain, 5 000 mornings. A rare condition leaves few trials to count in.
• Part D: not on its own: even groups of 50 ranged from 0.38 to 0.58, and 40 trials swing further, so 0.65 can happen when the true value is 0.5. Run more trials before judging the model.

More trials, steadier estimates

A. A running estimate

One long spreadsheet run gave 500 rainy mornings. Work out each estimate of P(late | rainy), plot it, and join the points. The first is done for you.

B. Groups of 10 and groups of 50

C. How many mornings?

D. Stretch: is the model wrong?

Design, run and report

Students put the whole cycle together: design a simulation for a player’s two free throws, run 30 trials from random digits, estimate a conditional probability both ways round, compare with the exact values, and report.

We are learning to

• design digit rules for a two-stage situation from a description,
• run and record trials, and estimate P(A | B) and P(B | A),
• compare estimates with exact values and report the limits of a simulation.

Success criteria

• I can write digit rules for the second throw that depend on the first.
• I can report how close 30 trials came and what would make the estimate better.

You need

Calculators; the worksheet; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; start the next with Part C.

Watch for these ideas

• One rule for both throws: the second throw’s rule depends on the first.
• Dividing by 30: P(second in | first in) divides by the first-in trials.
• Reporting a bare number: say how many trials, how close, and what would improve it.

Answers

• Part A: first throw 0 to 6 in; second throw after a make 0 to 7 in, after a miss 0 to 4 in.
• Part B: in then in 16, in then miss 5, miss then in 4, miss then miss 5; 21 first throws in.
• Part C: P(second in | first in) ≈ 16/21 ≈ 0.76 (exact 0.8); P(first in | second in) ≈ 16/20 ≈ 0.80. Exact: 0.7 × 0.8 = 0.56; with 0.3 × 0.5 the second goes in 0.71 of the time, so 0.56 ÷ 0.71 ≈ 0.79.
• Part D: a report names the model and rules, the method (30 trials), the estimates against the exact values, and a limit, such as only 21 trials inside the condition, or a real player’s chances changing.

Design, run and report: free throws

A basketball player takes two free throws. The first goes in 70% of the time. If the first goes in, the second goes in 80% of the time; if the first misses, the second goes in 50% of the time.

A. Design

B. Run thirty trials

Use the first digit of each pair for the first throw and the second digit for the second throw. Tally the results.

C. Estimate and compare

D. Stretch: report

Chances and the tools that make them

For Day 1, one set per pair. Six cards give a chance; six describe a fair tool. Match each chance to the tool that makes it.

Teacher note: the key is with the mini-check answers.

What goes on the bottom?

For Day 2, one set per pair. Each card asks for an estimate from simulated mornings. Sort the cards into three piles: divide by the rainy mornings, by the late mornings, or by all the mornings.

Teacher note: the key is with the mini-check answers.

What we know: Simulating conditional probability

Work on your own. A calculator is allowed. Show your working.

1. An event should happen with chance 0.3. Write a rule for it using random digits 0 to 9. ______________________
2. In 60 simulated mornings, 24 were rainy and 15 of those had a late bus. Estimate P(late | rainy). ______________________
3. In the same run, 21 mornings had a late bus. Estimate P(rainy | late). ______________________
4. Why is the estimate in question 2 less reliable than an estimate of P(rainy) from the same run? ______________________
5. Write a spreadsheet formula that gives “rain” with chance 0.4 and “dry” otherwise. ______________________
6. Two pairs run the same simulation and get 0.45 and 0.58. This shows, circle one:   one pair made a mistake  ·  the model is wrong  ·  estimates vary from run to run
7. A model gives P(late | rainy) = 0.5. After 400 rainy trials the estimate is 0.47. Is the model reasonable? Explain. ______________________
8. A student divides 15 by 60 to answer question 2. What did they estimate instead? ______________________

Answers and marking guide

Answers with working

1. Any 3 of the ten digits, such as 0, 1 and 2.
2. 15/24 ≈ 0.625: divide by the rainy mornings.
3. 15/21 ≈ 0.71: divide by the late mornings.
4. It rests on only 24 trials, the rainy ones, not all 60, so it swings more.
5. =IF(RAND()<0.4,"rain","dry")
6. It shows that estimates vary from run to run, as each run is a fresh sample.
7. Yes: 0.47 is close to 0.5 after 400 trials, and small differences are expected.
8. P(rainy and late), about 0.25, not the conditional.

A quick three-level guide

Eight questions, four ideas. A student at standard puts the condition’s count on the bottom and expects runs to differ.

Card keys

Chance cards (Day 1): 1 in 6 chance: A and G; 50% chance: J and D; 30% chance: L and E; 25% chance: H and B; 1 in 3 chance: F and K; 7% chance: C and I. Which-count cards (Day 2): the rainy mornings: M, P, R, V, W; the late mornings: O, Q, S, X; all the mornings: N, T, U.

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: a model you can run, counting inside the condition, the spreadsheet, more trials, and the report.