--- title: "Many trials" subtitle: "Data and chance · Chance · Lesson 8" --- # Many trials **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Flip a coin ten times and you might get 7 heads. Flip it ten thousand times and you will be very close to half. Why does more help? :::teacher-note **In one sentence:** Students run hundreds and thousands of trials and watch the relative frequency settle towards the probability (the law of large numbers), use large experiments to estimate unknown probabilities, and reject the idea that outcomes are 'due' (AC9M7P02, node 2). **Students should be able to say:** I can describe how the relative frequency changes as the number of trials grows. I can use a large experiment to estimate a probability. I can explain why a run of one outcome does not make another one 'due'. **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 Picture 10 flips. How likely is exactly 5 heads? :::keep-together **Which is more likely to give a relative frequency of heads between 0.45 and 0.55: 10 flips, or 1000 flips?** ☐ 10 flips ☐ 1000 flips ☐ Both are equally likely My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 1000 flips. In 10 flips, getting 3 or 7 heads (0.3 or 0.7) is common. Over 1000 flips the relative frequency is almost always within 0.05 of 0.5. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Settling, not balancing :::keep-together ### Settling, not balancing Here is why more trials help. 1. $\tfrac{7}{10} = 0.7$ — Ten flips: one result changes the fraction by 0.1. 2. $\tfrac{7}{10} \to \tfrac{8}{11} \approx 0.73$ — One more head moves it a lot. 3. $\tfrac{501}{1000} \to \tfrac{502}{1001} \approx 0.5015$ — After 1000 flips, one more head barely moves it. 4. $\text{each flip: } P(H) = \tfrac{1}{2}$ — The coin has no memory: nothing is "due". Now find the relative frequency after 1000 flips. **After 1000 flips, a coin has shown 482 heads. What is the relative frequency of heads? (A decimal.)** Answer: {{field-15}} ::: :::teacher-note **Answer:** 0.482 482 ÷ 1000 = 0.482: close to 0.5. ::: :::keep-together ### Find the mistake **Zara thinks about her next flip. Click the line where her reasoning goes wrong.** ``` 1 A fair coin has P(heads) = 1/2. 2 I flipped it 5 times and got 5 tails in a row. 3 Heads is now more likely on the next flip, to balance things out. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: the coin does not remember. Each flip is still 1/2 heads. Five tails in a row is unusual but not impossible, and it is not 'balanced out'; it just matters less as more flips are added. ::: :::keep-together ### Why it settles **Why does the relative frequency settle down over many trials?** ☐ The coin starts landing heads more to catch up ☐ Each new trial changes the total by less and less, so the fraction moves less ☐ After 1000 trials the outcomes stop being random ::: :::teacher-note **Answer:** Each new trial changes the total by less and less, so the fraction moves less One more head changes 7/10 a lot, but 501/1000 hardly at all. ::: :::typical ## One change at a time **A dice.** Each task changes one thing. **A mystery bag.** **Which to trust.** **Long runs.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Many trials :::keep-together ### Estimate **A drawing pin landed point-up 312 times in 800 throws. Estimate the probability it lands point-up. (A decimal.)** Answer: {{field-15}} ::: :::teacher-note **Answer:** 0.39 312 ÷ 800 = 0.39. A pin has no sample space of equal outcomes, so experiments are the way to estimate. ::: :::keep-together ### Streaks **A roulette wheel landed on red 8 times in a row. What is true about the next spin?** ☐ Black is now more likely ☐ Red is exactly as likely as before ☐ Red is now more likely: it is on a streak ::: :::teacher-note **Answer:** Red is exactly as likely as before Each spin is independent: the probability does not change. ::: :::keep-together ### Use the estimate **Using your estimate of 0.39, how many point-up landings would you expect in 200 throws?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 78 200 × 0.39 = 78. ::: :::keep-together ### Review: expected count **Review: how many heads do you expect in 300 flips of a fair coin?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 150 300 × 1/2 = 150. ::: ## Make: how many is enough? How many flips did it take? Compare with others. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **After 40 spins a spinner gave 18 reds; after 4000 spins, 1230 reds. Which estimate of P(red) is better, and what is it?** {{lines-3}} ::: :::teacher-note **Answer:** The 4000-spin one: 1230 ÷ 4000 ≈ 0.31. More trials give a more reliable estimate. :::