Repeated Chance Experiments

Expected heads: ___. Actual heads: ___

Difference from expected: ___

Is this result reasonable? Why? ___

Expected count for each number: ___

Which number appeared most? Is this expected? ___

Which number appeared least? Is this expected? ___

What will you test? ___

How many trials? ___

Possible outcomes: ___

Prediction: ___

Experiment description: ___

Expected results: ___

How will you record results? ___

Results (H or T for each flip): ___

Tally: Heads: ___ Tails: ___

Were results close to expected? ___

Tally for each number: 1:___ 2:___ 3:___ 4:___ 5:___ 6:___

Expected count for each: ___

Which number appeared most? Least? ___

Were results close to expected? ___

Who got closest to the expected result? ___

If we combine all results: Total H: ___ Total T: ___ Total flips: ___

Are the combined results closer to expected than individual results? ___

Expected for each number in 30 rolls: ___

Which number appeared most? Is this expected? ___

If rolled 300 times, would results be closer to expected? Why? ___

Expected for each section: ___

Biggest difference from expected: section ___ (off by ___)

Does this mean the spinner is unfair? Explain: ___

A coin was flipped 200 times: 108 heads. Experimental probability of heads = ___ (as a fraction and decimal)

A die was rolled 120 times. A 6 appeared 18 times. Experimental probability of 6 = ___

How does the experimental probability compare to the theoretical probability? ___

A coin gives 85 heads out of 100 flips. Fair or unfair? Explain: ___

A die gives each number 9-11 times out of 60 rolls. Fair or unfair? Explain: ___

Step 1: Describe the spinner: ___

Step 2: How many trials? ___

Step 3: What results would suggest it is fair? ___

Step 4: What results would suggest it is unfair? ___

Describe your experiment: ___

How many trials will you do? ___

What do you predict? ___

How will you know if results match expectations? ___

Expected for each colour in 60 spins: ___

Which colour appeared most? Is this unusual? ___

If spun 600 times, expect each colour about ___ times.

Expected red: ___. Expected blue: ___.

Was the result close to expected? ___

If drawn 200 times, expected red: ___. Expected blue: ___

Spinner description (how many sections and what colours): ___

Prediction for 30 spins: ___

How will you record results? (table, tally chart, etc.): ___

How will you know if the spinner is fair? ___

Tally: Heads: ___ Tails: ___

Expected: Heads: ___ Tails: ___

Difference from expected: ___

Would you expect results to be closer with 300 flips? Why? ___

Expected 6s in 30 rolls: ___

Whose results were closest to expected? ___

Combined 6s out of 120 rolls: ___. Expected: ___

Are combined results closer to expected? ___

Heads appeared 45 times in 100 flips. Relative frequency of heads = ___

A 6 appeared 8 times in 60 rolls. Relative frequency of 6 = ___ (as fraction and decimal)

Red appeared 15 times out of 50 spins. Relative frequency = ___

Expected: Red ___, Blue ___, Green ___

Was red close to expected? ___

Were blue and green close? ___

Do results support the claimed probabilities? Explain.

A drawing pin was tossed 50 times: point up 32 times. Experimental P(point up) = ___

Is a drawing pin a fair experiment? (Do both outcomes have equal probability?) ___

Based on 50 tosses, predict results for 200 tosses: point up ___, point down ___

A spinner gives: Red 45, Blue 35, Green 20 out of 100 spins. Are all sections equal? Explain.

A coin gives 490 heads out of 1000 flips. Is it fair? Why or why not?

How many trials would you need to be confident about whether a coin is fair?

You think a die might be weighted. Design an experiment to test this.

How many rolls would you need? ___

What results would make you confident the die is fair?

What results would make you think it's unfair?

Expected for each in 100 spins: ___

Which section appeared most above expected? ___. By how much? ___

Do these results suggest an unfair spinner? Explain: ___

A bag has 3 red, 4 blue and 3 green marbles. You draw one marble (and replace it). Prediction for 50 draws: Red: ___ Blue: ___ Green: ___

After 50 draws: Red: 17, Blue: 19, Green: 14. Were results close to expected? ___

Would 200 draws give closer results? Why? ___

Green appeared 18 times in 90 spins. Relative frequency = ___ (fraction, decimal, %)

A die showed 6 appeared 22 times in 120 rolls. Relative frequency = ___

As the number of trials increases, relative frequency gets closer to ___ probability.

Amy: 11H, Ben: 8H, Cara: 13H, Dan: 9H, Erin: 10H. Total H: ___ out of ___

Who was closest to expected? ___. Furthest from expected? ___

Combined relative frequency = ___ (close to 0.5?)

Does combining results give a more reliable estimate? ___

Heads in 50 flips: 28. Theoretical P(heads) = ___. Experimental P(heads) = ___

Are they close? Would 500 flips give a closer result? ___

If you ran the experiment 1000 times, which is more likely: exactly 500H or approximately 500H?

How many trials would you run? ___. Why?

What results would suggest the coin is fair? ___

What results would strongly suggest it is biased? ___

How would you record and display your results? ___

What is the difference between theoretical and experimental probability?

Give an example where you can calculate theoretical probability: ___

Give an example where you can only use experimental probability: ___

Which type of probability is more reliable for prediction? Explain.

Ali flips 10 times: gets 8 heads. Does Ali think the coin is fair? ___

Mia flips 200 times: gets 108 heads. Does Mia think the coin is fair? ___

Whose results are more reliable? Why?

How would you decide the coin is definitely biased?

Actual results: 1→8, 2→12, 3→9, 4→11, 5→7, 6→13. Write the relative frequency for each as a fraction.

Which face appeared most above expected? ___

Do these results suggest the die is biased? Explain.

Why is it useful to graph experimental results as relative frequencies instead of counts?

If you doubled the number of trials, how would the relative frequencies change?

Sketch what you'd expect a graph of relative frequency to look like after 10, 50, 100 and 500 trials for P(heads) = 0.5:

Expected for each in 30 spins: ___

Could this result happen by chance even if the spinner is fair? ___

What would you do next to find out if the spinner is biased?

A cereal company says their box always contains one of 6 toys equally. Design an experiment to test this:

How many boxes would you need to open? ___. What results would suggest the claim is correct? ___

What would the relative frequency for each toy be if the claim is correct? ___

Expected rolls for each of the 6 faces: ___

Actual results: 1→105, 2→98, 3→102, 4→96, 5→101, 6→98. Does the die seem fair? ___

Which face appeared most often? ___. Is this cause for concern? ___

After 100 coin flips, you get 53 heads. Relative frequency of heads: ___

Theoretical P(heads): ___. Are they the same? ___

After 10,000 flips, relative frequency would be closer to ___. Why?

If a baby has a 50% chance of being a boy, how could you use a coin to simulate this? ___

If a player makes 1 in 3 free throws, how could you use a die to simulate a game? ___

Run 10 simulated 'free throw' trials. Results: ___. Expected makes: ___

A pea plant can produce green or yellow peas, each with P = 1/2. In 40 plants, expected yellow: ___

Actual result: 23 yellow, 17 green. Close to expected? ___

Why might the actual result differ from the expected?

A spinner has 3 equal sections (A, B, C). After 30 spins: A=12, B=11, C=7. Calculate relative frequencies: A=___ B=___ C=___

Theoretical probability for each: ___. Are results close? ___

If the experiment is repeated 300 times, predict how many times C would appear: ___

A test correctly identifies a disease 95% of the time. P(correct result) = ___

If 200 people with the disease are tested, about how many correct identifications? ___

Why might even a 95% accurate test produce false positives/negatives in practice?

Choose an experiment: rolling a die or flipping a coin. Record 40 trials in a tally table:

Calculate the relative frequency for each outcome: ___

Compare your relative frequencies with the theoretical probabilities. What do you notice?

Misconception: 'I've rolled 5 tails in a row, so heads is now MORE likely.' Explain why this is wrong:

Misconception: 'The probability of getting at least one 6 in 6 rolls is 6/6 = 1.' Why is this wrong?

A drawing pin was dropped 100 times: 62 times it landed point up, 38 times point down. P(point up) ≈ ___

If dropped 500 times, expected point up: ___

Why can't we calculate a theoretical probability for a drawing pin? ___

Experiment: Does a thumbtack land point-up or point-down? How would you ensure fair results?

How many trials would give reliable results? ___. Why do more trials give better estimates?

How would you display your results? ___

Three students each rolled a die 30 times. Results for '6': Student A: 3, Student B: 8, Student C: 5. Who is closest to expected? ___

If you combined all three experiments (90 rolls), expected sixes: ___. Actual: ___

What does combining results from multiple experiments show?

Medical researchers test a new drug on 1,000 patients. Why do they need so many patients (not just 10)?

Quality control: a factory checks 200 out of 10,000 products and finds 4 defective. Estimate defective in total: ___

How could you improve the reliability of this estimate?