Coin Dynamics
Bernoulli trials at scale, from a single flip to one million.
Flip a virtual coin up to large sample sizes and observe how short-term streaks gradually converge toward the long-run 50/50 probability, a live demonstration of the law of large numbers.
Runs entirely in your browser, no server involved. Up to 1,000,000 flips processed in real time.
No flips this session.
Frequently Asked Questions
Is a coin flip truly 50/50? ▾
A perfectly fair coin flip is 50/50. However, research shows physical coins have a slight bias toward the side facing up when tossed, due to the dynamics of the flip. The effect is small (~51% vs 49%) and only detectable over thousands of flips.
What does the chi-square test show? ▾
The chi-square test measures whether the observed distribution of heads and tails differs significantly from the expected 50/50. A p-value above 0.05 means the results are consistent with a fair coin. A p-value below 0.05 suggests the distribution is unusually skewed.
Can you get 10 heads in a row with a fair coin? ▾
Yes. The probability of 10 consecutive heads is (0.5)^10 = 1/1024 ≈ 0.098%. Rare, but not remarkable. Over 10,000 flips you would expect this to happen roughly 10 times. Streaks feel meaningful but are a normal feature of random sequences.
What is the Law of Large Numbers? ▾
The Law of Large Numbers states that as the number of trials increases, the observed frequency converges toward the theoretical probability. Flip 10 coins and you might see 7 heads. Flip 1,000,000 and the proportion will be very close to 50%. Use the 1M flip option to see this in action.