Sampling Variability with an Unequal Spinner
Mr. Rivera's third graders are investigating a spinner with four sectors: red, blue, green, and yellow. The sectors are not equal: red is half the circle, blue is one quarter, and green and yellow are each one eighth. The class predicts that red should appear most often. In 20 spins, the results are red 9, blue 4, green 5, yellow 2. Aisha says, 'The spinner is not fair because green came up more than blue, but blue is supposed to be bigger.' Mr. Rivera wants students to separate theoretical probability from experimental results and to recognize sampling variability. He has the spinner, a tally sheet, and a computer with a spinner simulator. The spinner is mounted on cardboard, and the sectors are visibly unequal.
1. How should Mr. Rivera interpret Aisha's claim? Is her conclusion justified by 20 spins? Explain using theoretical probabilities and expected counts.
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Aisha is comparing one observed result with one theoretical prediction and treating a small difference as proof of unfairness. The theoretical probabilities are red 1/2, blue 1/4, green 1/8, and yellow 1/8. In 20 spins, the expected counts are red 10, blue 5, green 2.5, and yellow 2.5. The observed counts are red 9, blue 4, green 5, and yellow 2. Green is above its expected count and blue is below, but with only 20 spins, that pattern can easily occur by chance even if the spinner is fair. Her conclusion is therefore not justified. A single sample of 20 does not establish unfairness. To investigate fairness, the class should look at the whole distribution and collect more data, not just compare green and blue once. The key idea is sampling variability: experimental results vary around theoretical probabilities, and small samples can produce misleading proportions.
2. What question or activity would help Aisha and her classmates distinguish 'what should happen' from 'what did happen'? Include specific numbers.
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Ask students to predict expected counts for 80 spins: red 40, blue 20, green 10, and yellow 10. Then have them run many simulated sets of 20 spins and record how often green exceeds blue even though blue is theoretically more likely. This directly tests whether Aisha's 20-spin result is surprising. Another activity is to pool the class's data to 200 spins and compare the observed counts with expected counts of red 100, blue 50, green 25, and yellow 25. Ask, 'If we spun 20 times again, would we necessarily get exactly 10 red, 5 blue, 2.5 green, and 2.5 yellow?' Students should recognize that expected counts are long-run averages, not guarantees for a single set of spins. The simulator lets them see a dot plot of many 20-spin experiments, so they can observe that green occasionally beats blue even with a fair spinner. That experience helps them separate the theoretical model from any one experimental result.
3. A student says, 'If we spin 1,000 times, we will get exactly 500 reds.' Evaluate this statement and explain what the Law of Large Numbers does and does not guarantee.
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The statement is false if it means exactly 500 reds. With red probability 1/2, the expected count in 1,000 spins is 500, but the actual count will vary. A result such as 480 or 520 reds would not be surprising. The Law of Large Numbers says that as the number of spins increases, the proportion of reds tends to get closer to 1/2; it does not guarantee an exact count in any fixed number of trials. It also does not mean the spinner 'remembers' earlier results and compensates. After 9 reds in 20 spins, the spinner is not 'due' for fewer reds; each spin still has probability 1/2 for red if the spinner is fair. The distinction matters for interpreting data: larger samples tend to give more stable relative frequencies, but they still show variability. Students should compare proportions, not demand exact expected counts, and should recognize that probability models describe long-run behavior rather than deterministic outcomes.