--- title: "Simulating compound events" subtitle: "Data and chance · Chance · Lesson 14" --- # Simulating compound events **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** In a board game you roll two dice every turn. Which total should you hope for, and how sure can you be? Rolling by hand would take all lesson; a simulation takes a second. :::teacher-note **In one sentence:** Students simulate two dice and two coins, record results in a frequency table, and compare relative frequencies with probabilities from the sample space (AC9M8P03, nodes 1 and 2). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict :::keep-together **Two dice are rolled 100 times and the totals recorded. Which total will probably come up most often?** ☐ They will all come up about the same ☐ 7 ☐ 12 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 7. More pairs make 7 (six of the 36) than any other total; 2 and 12 have just one pair each. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Reading a simulation :::keep-together ### Relative frequency Here is the routine. 1. $\text{Event: total of 9};\ P = \tfrac{4}{36} \approx 0.11$ — Predict from the grid first. 2. $\text{Run } 500 \text{ trials: } 9 \text{ came up } 58 \text{ times}$ — Simulate and record. 3. $58 \div 500 = 0.116$ — Relative frequency. 4. $0.116 \approx 0.11$ — Close: more trials would make it closer still. **In 200 simulated rolls of two dice, a total of 7 came up 31 times. What is its relative frequency?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 0.155 31 ÷ 200 = 0.155, close to 1/6 ≈ 0.167. ::: :::keep-together ### Find the mistake **Ella runs a short simulation. Click the line where her conclusion goes wrong.** ``` 1 Ella uses an app to roll two dice 20 times. 2 A sum of 12 never comes up in her 20 rolls. 3 So the probability of a sum of 12 is 0. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** 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. ::: :::keep-together ### 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 ☐ =RANDBETWEEN(1,12) ☐ =A2+B2 ☐ =RANDBETWEEN(2,12) ::: :::teacher-note **Answer:** =A2+B2 Add the two dice. One random number from 2 to 12 would make every total equally likely, which two dice don't. ::: :::typical ## More trials, other events **1000 trials.** **At least one head.** **A and not A.** **Expected count.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Simulations :::keep-together ### 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.** {{rows 1}} | outcome | HH | HT | TH | TT | |---|---|---|---|---| | count | | | | | ::: :::teacher-note **Answer:** 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. ::: :::keep-together ### 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 A: its relative frequency (0.225) is bigger ☐ Class B ☐ They are equally good ☐ Neither: simulations don't work ::: :::teacher-note **Answer:** Class B More trials: 0.17 from 1000 trials is far steadier than 0.225 from 40. ::: :::keep-together ### 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 ☐ The simulation is broken ☐ The probability is wrong ☐ The probability is really 0.31 ::: :::teacher-note **Answer:** They agree well enough They are close: the difference is chance. Relative frequency varies, and settles near the probability as the trials grow. ::: :::keep-together ### Review: or **Review: 30 students, 18 play netball, 10 play hockey, 4 play both. What is P(netball or hockey)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 24/30 18 + 10 − 4 = 24, so 24/30 = 4/5. ::: ## Make: your own simulation Describe your result in a sentence: the event, its probability, the relative frequency after your trials, and whether they agree. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **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.** {{lines-3}} ::: :::teacher-note **Answer:** 68 ÷ 500 = 0.136. P(8) = 5/36 ≈ 0.139: very close. :::