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Empirical Law of Large Numbers: Coin Toss Frequency Exploration

Empirical Law of Large Numbers: Coin Toss Frequency Exploration
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2026-01-01 18:16
SequenceLimitProbability & StatisticsSlider AnimationGraphing
In collections Probability and Statistics
① This example simulates tossing a fair coin many times and dynamically shows how the frequencies of heads and tails converge to the theoretical probability of 0.5 as the number of trials increases, illustrating the empirical law of large numbers. ② Core mathematical concepts include the law of large numbers, the relationship between frequency and probability, random experiments, approximation to binomial distribution, and the idea of limits. ③ Construction highlights: using GeoGebra commands Sequence to generate trial sequences, Take to extract the first n results, CountIf to count heads/tails, and BarChart to plot frequency bars; key objects include n (trial count slider), lista1 (random result list), uscite (outcomes), teste/croci (head/tail counts), FreqTeste/FreqCroci (frequencies), PercTeste/PercCroci (percentages), linProbTeo (theoretical probability line), along with buttons and option interactions. ④ Suitable for high school and above in probability and statistics, or introductory university probability courses, to help understand frequency stability and randomness.

GeoGebra Commands & Parameters

  1. Sequence(Random[0, 1], k, 1, 5000) → lista1, {0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 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0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1}
  2. Take(lista1, 1, n) → uscite
  3. CountIf(x ≟ 0, uscite) → teste
  4. CountIf(x ≟ 1, uscite) → croci
  5. BarChart(0, 2, {FreqTeste, FreqCroci}) → a

AI Drawing Prompt Reference

Simulate tossing a fair coin many times and dynamically show how the frequencies of heads and tails converge to the theoretical probability of 0.5 as the number of trials increases, illustrating the law of large numbers. Display the number of heads, tails, frequency, percentage, and theoretical probability line.
The prompt above was compiled from the original drawing commands and is for reference only.