The Spinner Game Argument
A spinner has 8 equal sections: 4 sections are red, 2 are blue, 1 is green, and 1 is yellow. In a game, a player wins a point if the spinner lands on red. Priya spins 20 times and records: red 9, blue 6, green 3, yellow 2. She says, "Blue came up more than green, so blue must be more likely than green - but green came up 3 times and I expected about 2.5, so the spinner is broken." The class debates whether 20 spins is enough to judge the spinner. They also compare Priya's results to what the sections suggest.
1. Using the sections, state the probability of each colour as a fraction of 8. Then compare the expected number of each colour in 20 spins with Priya's results, and judge whether her results prove the spinner is unfair.
Read the worked answer
Probabilities from the sections (8 equal parts):
- Red: 4/8 = 1/2
- Blue: 2/8 = 1/4
- Green: 1/8
- Yellow: 1/8
Expected counts in 20 spins:
- Red: 1/2 × 20 = 10
- Blue: 1/4 × 20 = 5
- Green: 1/8 × 20 = 2.5
- Yellow: 1/8 × 20 = 2.5
Compare with Priya:
- Red 9 (expected 10)
- Blue 6 (expected 5)
- Green 3 (expected 2.5)
- Yellow 2 (expected 2.5)
Every result is close to what the sections predict. Small differences are normal in a random experiment, and 20 spins is a small sample, so the results do not prove the spinner is unfair. Priya's claim that blue "must be more likely" than green is actually correct from the sections (2/8 vs 1/8), so her results agree with the design rather than contradict it.
2. Priya wants to be more confident about whether the spinner is fair. Describe a better plan and explain how the expected counts would change if she used it.
Read the worked answer
A better plan is to spin many more times, for example 200 spins, and compare results with expected counts.
Expected counts in 200 spins:
- Red: 1/2 × 200 = 100
- Blue: 1/4 × 200 = 50
- Green: 1/8 × 200 = 25
- Yellow: 1/8 × 200 = 25
With more spins, the results should get closer to these expected numbers. A difference of, say, 4 or 5 from the expected count is a small fraction of 100, but the same difference is large compared with 2.5, which is why 20 spins is unreliable.
She should also record results in a tally or table and repeat the whole experiment if results look far off. Only large, repeated gaps between observed and expected counts would be good evidence that the spinner is unfair.