--- title: "Predict and test" subtitle: "Data and chance · Chance · Lesson 7" --- # Predict and test **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Roll a dice 60 times. Exactly ten sixes? Probably not. But how far off is normal, and when should you suspect the dice? :::teacher-note **In one sentence:** Students predict expected counts (probability × trials), run experiments on a chance board, record counts and relative frequencies, and see that observed results vary around the expected count (AC9M7P01 node 3, AC9M7P02 node 1). **Students should be able to say:** I can predict how often an outcome should happen in a number of trials. I can run an experiment, record counts and work out relative frequencies. I can explain why results differ from predictions in a short experiment. **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 What fraction of rolls should be a six? :::keep-together **You roll a fair dice 60 times. About how many sixes do you expect?** ☐ About 30 ☐ You can't say anything ☐ About 10 ☐ Exactly 6 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** About 10: 60 × 1/6 = 10. The real count will usually be near 10, but not always exactly 10. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Expected and observed :::keep-together ### Expected, then observed Here is how to predict and test. 1. $P(6) = \tfrac{1}{6}$ — The probability from the sample space. 2. $60 \times \tfrac{1}{6} = 10$ — Expected count: probability × trials. 3. $\text{observed: } 7 \text{ sixes}$ — Run the experiment and record the count. 4. $\tfrac{7}{60} \approx 0.12$ — Relative frequency = count ÷ trials, the experiment's estimate. 5. $0.12 \text{ against } 0.17$ — Close but not equal: a short experiment varies. Now: how many heads in 50 flips? **You flip a fair coin 50 times. How many heads do you expect?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 25 50 × 1/2 = 25. ::: :::keep-together ### Find the mistake **Noah tests a dice. Click the line where his reasoning goes wrong.** ``` 1 I rolled a dice 60 times. 2 P(6) = 1/6, so I expect 60 × 1/6 = 10 sixes. 3 I got 7 sixes, not 10. 4 So the dice must be unfair. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 4 Line 4: in 60 rolls the count of sixes often lands a few away from 10; 7 is common for a fair dice. The expected count is what happens on average, not every time. To test a dice, roll it many more times. ::: :::keep-together ### Relative frequency **In an experiment, a coin landed heads 13 times in 20 flips. What is the relative frequency of heads? (A decimal.)** Answer: {{field-15}} ::: :::teacher-note **Answer:** 0.65 13 ÷ 20 = 0.65, more than the probability 0.5: chance varies in 20 flips. ::: :::typical ## One change at a time **Flip 100.** Each task changes one thing. **An unequal spinner.** **Test the spinner.** **Two results.** **Not a whole number.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Expected counts :::keep-together ### Expected count **A bag has 2 red and 8 blue counters. You take one and put it back, 40 times. How many reds do you expect?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 8 P(red) = 2/10 = 1/5, and 40 × 1/5 = 8. ::: :::keep-together ### Relative frequency **A spinner landed on green 18 times in 60 spins. What is the relative frequency of green? (A decimal.)** Answer: {{field-15}} ::: :::teacher-note **Answer:** 0.3 18 ÷ 60 = 0.3. ::: :::keep-together ### Suspicious **Which result would make you most suspicious of a coin?** ☐ 52 heads in 100 flips ☐ 6 heads in 10 flips ☐ 900 heads in 1000 flips ::: :::teacher-note **Answer:** 900 heads in 1000 flips Over 1000 flips you expect about 500; 900 is far too many for chance. ::: :::keep-together ### Review: probability **Review: roll a fair dice. What is $P(\text{a number less than } 3)$? Simplest form.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1/3 {1, 2}: 2/6 = 1/3. ::: ## Make: plan an experiment Then: how many spins of the 3-red, 5-blue spinner to expect 30 reds? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A spinner has P(blue) = 0.4. How many blues do you expect in 50 spins? You get 23. Is the spinner unfair?** {{lines-3}} ::: :::teacher-note **Answer:** 20 expected. 23 is close to 20, so there is no reason to think it is unfair. :::