Math // Probability

Experimental vs Theoretical

Run repeated probability experiments and compare experimental (relative frequency) results with theoretical predictions; understand and demonstrate that as the number of trials increases, the experimental probability tends towards the theoretical probability — and that short runs can give very different results

Ages
10 to 11
Type
Method
Builds on
2 topics
Leads to
1 topic

Quick check

If the learner flips a coin 10 times and gets 7 heads, do they understand why this doesn't mean heads is "more likely" — and that the more times you flip, the closer the results get to 50:50?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Roll a die 60 times and compare experimental frequencies with the expected 10 per number
  • Explain why experimental results don't exactly match theoretical predictions but get closer with more trials
  • Predict what would happen if the experiment were repeated 600 times instead of 60

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics
  • Simple Chance ExperimentsRequired

    Probability // Ages 9 to 10

    Comparing experimental and theoretical probability requires prior experience conducting simple experiments and recording results

  • Calculating Simple ProbabilityRequired

    Probability // Ages 10 to 11

    Comparing experimental results with theoretical predictions requires being able to calculate the theoretical probability first

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Experimental vs Theoretical

Ages 10 to 11 // Method

Leads to

1 topic
  • Experimental probabilityRequired

    Probability // Ages 11 to 13

    The formal KS3 frequency analysis of experimental probability extends the bridging work on long-run behaviour introduced at age 10-11

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Learners compare experimental probability (relative frequency measured from actual trials) with theoretical probability (calculated from equally-likely outcomes). The central idea is a child-level law of large numbers: as the number of trials grows, the experimental relative frequency tends toward the theoretical value, while short runs can differ widely and each trial stays independent. Learners run trials, compute relative frequency from their OWN tally (frequency ÷ number of trials), and compare with prediction — and they confront two common fallacies: that experimental results should equal theory exactly, and that outcomes become 'due' or that absolute counts catch up to be equal.

Common misconceptions

  • Expecting experimental results to equal the theoretical probability, and calling a device biased after a short run (law of small numbers)
  • Thinking absolute counts 'catch up' to become exactly equal with more trials (gambler's fallacy of convergence / the 'due' idea)
  • Reporting the theoretical probability when asked for the experimental one (or vice versa)
  • Trusting a small sample as much as a large one to estimate an unknown probability

Useful representations

  • Tally / frequency table with a relative-frequency column (f ÷ n)
  • Relative-frequency line converging toward the theoretical line as trials increase
  • Short-run vs long-run comparison side by side
  • Experimental value and theoretical value marked on the same 0-to-1 scale

Teaching practices

  • Have children compute relative frequency from THEIR OWN tally (frequency ÷ trials), not restate the theoretical value
  • Contrast a short run with a long run — pool class data or use a simulator — so convergence toward theory is seen, not asserted
  • Name explicitly that closeness improves with MORE trials but exact equality is not expected
  • Attack the 'due' fallacy directly: the die does not self-correct; relative frequency converges, absolute counts do not equalise exactly
  • Use the comparison to judge fairness sensibly — a big deviation over many trials suggests bias, a deviation over a few does not

Activities to try

  1. 1

    Roll, tally, and compute your own experimental probability

    Why it works: Experimental probability comes from the data you collected (f ÷ n); it is usually close to but not exactly the theoretical value, and ordinary short-run differences are expected rather than proof of bias.

    1. Roll a die 60 times (or flip a coin / spin a spinner), tallying each result in a frequency table.
    2. Add a relative-frequency column: for each outcome compute frequency ÷ 60 as a fraction or decimal.
    3. State the theoretical probability (1/6 for a fair die) and mark both the experimental and theoretical values on the same 0-to-1 scale.
    4. Write in your own words how your experimental values compare with 1/6, and whether the differences you see are enough to call the die unfair.
  2. 2

    Short run vs long run: watch the proportion settle while counts stay unequal

    Why it works: It is the PROPORTION (relative frequency) that converges toward the theoretical value with more trials — not the absolute counts becoming equal. No outcome is ever 'due', and each trial is independent of the ones before it.

    1. Compare your own short run (60 rolls) with a much longer run — pool the whole class's tallies, or use a trials simulator set to 60, then 600, then 6000.
    2. At each trial count, record both the relative frequency for each outcome AND the raw counts.
    3. Predict before increasing the trials: 'If I roll 600 times instead of 60, will each number come up exactly 100 times?' Then check against the data.
    4. Describe what gets closer to what: check whether, at the largest trial count, the relative frequencies for a fair die all sit near 1/6 (or near the theoretical value for your coin or spinner), while the raw counts (for example 995 vs 1020) are still unequal.
  3. 3

    Judge a fairness claim from sample size

    Why it works: A small sample is not as trustworthy as a large one for estimating an unknown probability; a single ordinary deviation over few trials is not evidence of bias, and judging fairness sensibly depends on the number of trials behind the data.

    1. Read a scenario: 'A 4 came up 14 times in 60 rolls of a die.' Decide whether this proves the die is biased, and explain your reasoning.
    2. Decide whose data you would trust more to estimate the chance of an outcome for a possibly-bent coin: 20 flips or 500 flips, and say why.
    3. Order several sample sizes from least to most reliable for estimating an unknown probability.
    4. Explain how you WOULD decide, with much more confidence, whether the die is loaded — and why even many trials give strong evidence rather than absolute certainty.

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