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Investigation Questions
Use these alongside the tool above. Run experiments with coins, dice, spinners, two dice and bags. Use the Table, Bar Chart, Convergence and Sample Space tabs, and take Snapshots to compare results.
1
Select the Coin experiment. Press ▶ Run 1 and watch the animation. What do you see? What colour is the coin when it lands on Heads? What about Tails? Press Run 1 five more times using the Space bar. Look at the Log tab — how many Heads and how many Tails did you get? Were you surprised?
2
Switch to the Table tab. You should see two rows: Heads and Tails. The Frequency column shows your counts. The other three columns are hidden behind hatched patterns. Click the Relative Freq 👁 header to reveal the whole column. What do the numbers mean? How do they relate to the frequencies?
3
Still on the Coin, press ×1000 to run a thousand trials at once. Now look at the Table again. How have the relative frequencies changed? Are they closer to 0.5 than before? Press ×1000 two more times. What is happening to the relative frequencies as you add more trials?
4
Switch to the Convergence tab. You should see two lines wobbling and then settling. Press 👁 Show Theoretical to overlay the dashed target lines. Where are the theoretical lines? Are the solid lines close to them? Now press ×10 several more times and watch the lines tighten. What mathematical law is this demonstrating?
5
Switch to the Bar Chart tab. You should see two bars. Toggle 👁 Show Theoretical to see dashed lines. Are the bars above or below the theoretical lines? Now hide them again. A student looks at your bar chart and says ‘This coin is unfair because the bars aren’t exactly equal.’ Do you agree? Explain your reasoning.
6
In the Coin config panel, change Number of Coins to 2. Reset the results (double-tap 🗑). The outcomes are now ‘0 Heads’, ‘1 Head’ and ‘2 Heads’. Run 100 trials. Which outcome is most common? Which is least common? Why isn’t ‘1 Head’ equally likely as the others?
7
Keep 2 coins. Run 1000 trials and look at the Table. Reveal the Theoretical column. The probability of 1 Head should be 0.5000, while 0 Heads and 2 Heads are each 0.2500. Can you explain these numbers? How many equally likely outcomes are there when flipping 2 coins, and what are they?
8
Change to 3 coins. Before running any trials, predict: which outcome will be most common — 0, 1, 2 or 3 Heads? Now run 1000 trials and check. Reveal the Theoretical column. Why is 0 Heads less likely than 1 Head?
9
Now change to 5 coins. Run 2000 trials and look at the Bar Chart. Describe the shape. Where is the peak? Is the distribution symmetric? Press 👁 Show Theoretical to overlay the expected values. How closely do your bars match?
10
Go back to 1 coin. Move the Coin Bias slider to P(Heads) = 0.80. Run 500 trials. Look at the Bar Chart. How does it compare to a fair coin? Now take a 📌 Snapshot. Change the bias back to 0.50 and run 500 more trials. The ghost bars from the snapshot appear behind your live bars. What differences can you see? Which bar chart looks ‘fair’?
11
With 1 coin at P(Heads) = 0.70, run only 10 trials. Look at your results. Can you tell this coin is biased from just 10 trials? Now run 1000 trials. Can you tell now? What does this teach you about sample size and detecting bias?
12
Switch to the Dice experiment with a d6. Run 60 trials. Switch to the Table tab and reveal the Expected 👁 column. Every row should show 10.0. Why? Now look at your actual frequencies. Which outcome was furthest from 10? Does this mean the die is unfair?
13
With d6, run 600 trials. On the Table, reveal all columns. Compare Frequency to Expected for each outcome. Calculate the difference. Now run 6000 trials and compare again. Are the differences smaller or larger? Are they a smaller or larger proportion of the total?
14
Take a 📌 Snapshot of your d6 results (after at least 500 trials). Now switch to a d8 and run the same number of trials. Look at the Bar Chart — the snapshot ghost bars show d6 alongside your d8 data. What are the key visual differences? Which has taller bars? Why?
15
Try a d20. Run 1000 trials and look at the Bar Chart. With 20 equally likely outcomes, what should each bar’s relative frequency be? Reveal the Theoretical column to check. Now look at the Convergence graph — why is it so much messier than the coin convergence graph?
16
A student claims: ‘A d4 is more predictable than a d20 because it has fewer outcomes.’ Run 200 trials on each and compare the Convergence graphs. Which one’s lines settle faster? Does ‘fewer outcomes’ mean ‘more predictable’? What exactly does predictable mean here?
17
Switch to the Spinner with 4 equal sectors. Run 400 trials. What should the theoretical probability of each sector be? Reveal the Theoretical column to check. How close are your relative frequencies?
18
Now use the +/− buttons on the sector weights to set the weights to 1 : 1 : 1 : 3 (increase Sector 4 to weight 3). Before running any trials, predict: what fraction of trials should land on Sector 4? Write down your prediction, then run 600 trials and check. Were you right?
19
With weights 1:1:1:3, take a 📌 Snapshot. Now press Equal to reset all weights to 1:1:1:1, and run 600 trials. Compare the snapshot ghost bars with your live bars. Which sectors changed the most? Write a sentence explaining what the snapshot comparison shows.
20
Set up a 2-sector spinner with weights 1 : 1 (a fair spinner). Run 500 trials. Now change the weights to 1 : 2 and run 500 more trials. Then try 1 : 3, 1 : 5, and 1 : 9. For each, note the relative frequency of Sector 1. As the spinner becomes more biased, what happens to P(Sector 1)? What weight ratio would make Sector 1 almost impossible?
21
Can you set up a spinner where one sector has probability exactly ⅓? What about exactly ¼? What about exactly ⅗? Experiment with different numbers of sectors and weights. Is every fraction achievable?
22
Switch to Two Dice with two d6 in Add mode. Before running any trials, switch to the Sample Space tab. The grid is blank. Now run single trials using ▶ Run 1 and watch cells light up. After 20 trials, how many of the 36 cells have been discovered? Which cells are still blank? Can you predict which cells will be hardest to fill?
23
Run 200 trials on two d6 (add mode). Look at the Sample Space — check the progress counter. Then press 👁 Reveal All. Study the complete grid. Why does 7 appear in a diagonal line across the grid? Count: how many cells show 7? How many show 2? How many show 12? How does this explain why 7 is the most common sum?
24
With the Sample Space revealed, look at the legend below the grid. It shows counts like ‘7: 6/36’. This means 6 out of 36 equally likely outcomes give a sum of 7. Write this as a simplified fraction. Now do the same for the sum of 2, 6, and 11. What pattern connects the number of ways to make each sum?
25
Switch to Subtract mode (|Die 1 − Die 2|). Reset and look at the Sample Space. Run trials to discover cells, then Reveal All. Where is the difference of 0 on the grid? Why does 0 appear along the main diagonal? What is the most common difference? What is the least common?
26
Take a 📌 Snapshot of two d6 in Add mode (after 1000 trials). Switch to Subtract mode and run 1000 trials. Look at the Bar Chart with the snapshot ghost bars. The shapes are completely different. Describe the differences. Which distribution is more spread out?
27
Switch to two d4 in Add mode. The Sample Space is now a 4×4 grid with 16 cells. Reveal All. What is the most common sum? Now try two d8 — an 8×8 grid with 64 cells. What is the most common sum? Can you state a general rule: for two dN dice added together, what is the most common sum?
28
Switch to the Bag experiment. The default bag has 3 Red, 3 Blue and 3 Green counters. Run 90 trials and look at the Table. Reveal Expected — each colour should show 30.0. Why? Now add 3 more Red counters (press + Red three times) and run 90 trials. How have the expected values changed?
29
Set up a bag with 5 Red and 1 Blue. Before running trials, predict: what fraction of draws should be Red? Run 600 trials and check. Now add 1 more Blue (5 Red, 2 Blue). Run 600 trials. How did the relative frequencies change? Keep adding Blue counters one at a time, running 600 trials each time. At what point do Red and Blue become equally likely?
30
With 3 Red, 3 Blue and 3 Green (with replacement), run 100 trials. Now switch to Without replacement and run 100 trials. Look at the Log tab. In without-replacement mode, you’ll see ‘(N left)’ and ‘(refilled, N left)’ notes. Watch the bag visual as you press Run 1 — counters disappear! When the bag empties, what happens?
31
Set up a bag with 1 Red and 9 Blue, without replacement. Run single trials and watch. How many draws does it take before you get the Red? Reset and try again. Is it always the same? Run this experiment 20 times (resetting each time) and tally how many draws it takes to find Red. What is the average? What should the average be theoretically?
32
A bag has 4 Red and 4 Blue counters. A student says: ‘It doesn’t matter whether I use replacement or not — Red and Blue are equally likely either way.’ Are they right? Set up both modes and run 1000 trials each. Compare the relative frequencies. Now change to 4 Red and 1 Blue and test both modes. Is the student still right? When does replacement vs without replacement make the biggest difference?
33
Choose any experiment. Run 5 trials and note the relative frequency of one outcome. Now run 50, 500 and 5000 trials, noting the relative frequency each time. On the Convergence tab (with theoretical lines shown), describe what you see. Write a sentence explaining the Law of Large Numbers in your own words.
34
With a fair d6, run exactly 6 trials. Did you get one of each outcome? Reset and try again. How many times out of 10 resets do you get a ‘perfect’ set of 6 trials? Now run 6000 trials. The relative frequencies should all be close to 1/6 — but are any of them exactly 1/6? What does this tell you about the difference between theoretical and experimental probability?
35
Set up a fair coin. Run 10 trials and take a 📌 Snapshot. Run 100 more and take a new Snapshot (overwriting the first). Run 1000 more and compare your current convergence with the snapshot. Which settled faster? The short experiment or the long one? Export the data as CSV and open it in a spreadsheet. At which trial number did the relative frequency first stay within 0.02 of 0.5?
36
Run 500 trials on a fair coin. Take a 📌 Snapshot. Change the bias to P(Heads) = 0.60 and run 500 trials. On the Bar Chart, the ghost bars (snapshot) show the fair coin, and the solid bars show the biased coin. Can you see the difference? Now try P = 0.55. Is the difference still visible? What is the smallest bias you can detect from 500 trials?
37
On a fair d6, run 1000 trials and take a 📌 Snapshot. Switch to a d8 and run 1000 trials. The Table now shows a Snapshot column. For outcomes 1–6, compare the current relative frequencies with the snapshot. Both dice are fair, so why are the current values smaller than the snapshot values?
38
Set up a 4-sector spinner with equal weights. Run 500 trials and take a Snapshot. Now set weights to 1:1:1:3 and run 500 trials. Look at the Convergence tab. The solid lines show the biased spinner converging to different values for each sector. The dashed purple lines show the snapshot’s equal convergence. Use 👁 Show Theoretical as well. You now have three layers of information on one graph. Describe what each layer tells you.
39
On the Table tab, leave the Relative Freq, Expected and Theoretical columns all hidden. Run 100 trials on a d6. Look at the Frequency column only. Before clicking anything, write down your prediction for each hidden column. Then click to reveal one column at a time. How accurate were your predictions?
40
A student says: ‘If I flip a fair coin 100 times, I will get exactly 50 Heads.’ Another says: ‘I will get about 50 Heads.’ Run 100 trials on a fair coin ten times (resetting between each). Record the number of Heads each time. What was the minimum? The maximum? Which student is closer to right?
41
Two d6 in Add mode: a student says ‘I need to roll a 6 for my board game. That’s easy — 6 is average.’ Another says ‘You should hope for a 7 — that’s the most common.’ Use the Sample Space (Reveal All) to count how many ways to make 6 vs 7. Who is right? What is the probability of each?
42
Set up a bag with 3 Red and 2 Blue. What is P(Red)? Write it as a fraction, a decimal and a percentage. Now run 1000 trials. How close is your experimental relative frequency to the theoretical value? Multiply your relative frequency by 1000 — how does this compare to your actual Red count?
43
Is it possible to set up an experiment where every outcome has the same theoretical probability? Try it with each experiment type: Coin (1 coin, fair), Dice (any), Spinner (equal weights), Two Dice, Bag (equal counters). Which experiment types can achieve equal probability for all outcomes? Which cannot? Why?
44
The Gambler’s Fallacy: flip a fair coin 10 times and get 8 Heads. Is the next flip more likely to be Tails? Set up a coin and run 1000 trials. Look at the Log — find a run of 5+ Heads in a row. What came next? Does the coin ‘remember’ what happened before? Run the experiment several times and collect evidence for or against the Gambler’s Fallacy.
45
Using two d6 in Add mode, which sum from 2–12 takes the longest to appear? Run single trials until every sum has appeared at least once. How many trials did it take? Reset and try again. Is it always the same number? Can you explain why certain sums take longer to appear?
46
Design a ‘game’ using the spinner. Create an unfair spinner where Sector 1 wins a prize but has only a 1 in 10 chance. Set the weights to achieve this. Run 1000 trials to verify the probability. Now change it so the chance is exactly 1 in 4. Is there more than one way to set the weights? Can you achieve any target probability?
47
Use 📊 Export CSV to download your raw data from a 1000-trial d6 experiment. Open it in a spreadsheet. The Rel Freq columns show the cumulative relative frequency after each trial. Plot a line graph of one outcome’s relative frequency against trial number. Describe the shape. What does the graph look like for the first 10 trials? The first 100? All 1000?
48
The Birthday Paradox involves surprisingly high probabilities from repeated trials. Set up a bag with counters representing days: use the maximum 15 counters in as many colours as possible. Draw without replacement. How quickly do you get a repeated colour? This is a simplified model — but what does it show about coincidences in small samples?
49
Challenge: using any combination of experiments and the 📌 Snapshot feature, design a demonstration that clearly shows the difference between theoretical and experimental probability. Your demonstration should convince a sceptical student that: (a) experimental results are unpredictable in the short run, and (b) they become predictable in the long run. Write instructions so another student can replicate your demonstration.
50
Design your own Probability Simulator investigation. Choose a mathematical focus (fairness, sample size, convergence, comparing distributions, or something else), decide which experiments and features to use, and write three questions that another student could explore. Test them yourself first, then exchange with a partner. Which of your questions led to the most interesting discoveries?