Question

The theoretical probability of getting heads is 1/2 (50%). The experimental probability is the fraction of heads that are observed in actual flips and should be close to 1/2, tending toward 1/2 as the number of trials increases.

Answer 1

Your statement is correct.

More detail and why it’s true:
- Theoretical probability for a fair coin: P(heads) = 1/2.
- Experimental probability (empirical proportion) = number of heads observed / number of flips.
- By the Law of Large Numbers, the experimental proportion tends to the theoretical probability 1/2 as the number of flips n → ∞.

How close you should expect to be (typical fluctuations):
- For n independent fair flips, the number of heads is Binomial(n, 1/2). The standard deviation of the proportion p̂ = (#heads)/n is
sd(p̂) = sqrt(p(1−p)/n) = sqrt(0.25/n) = 0.5 / sqrt(n).
- Typical (one‑sigma) fluctuations are about 0.5/√n. For example:
- n = 10 → sd ≈ 0.158 (≈15.8%), so large swings are common.
- n = 100 → sd ≈ 0.05 (5%).
- n = 1,000 → sd ≈ 0.0158 (1.58%).
- For an approximate 95% confidence interval around 0.5 use ±1.96·sd. Example: with n = 10,000, 95% interval ≈ 0.5 ± 0.0098 (within about 1%).

How many flips to be within 1% (95% confidence)?
- Solve 1.96·(0.5/√n) ≤ 0.01 ⇒ n ≥ (1.96^2·0.25)/(0.01^2) ≈ 9,604 flips.

In short: experimental frequency should be close to 50%, but the closeness depends on sample size; larger n gives closer agreement with the theoretical 1/2.