A casual scattered group of generic copper coins on ivory linen. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Fairness specifies a probability, not a pattern
A fair independent coin has a heads probability of one-half on every flip. It is not required to alternate, avoid long runs or finish each group of ten with five heads. A generator that deliberately switches sides to correct an imbalance would create dependence and would not implement the usual independent-coin model.
Quantify a common short-batch expectation
In ten independent fair flips, the probability of exactly five heads is C(10,5)/2¹⁰ = 252/1,024, or 24.609375%. Thus more than three-quarters of ten-flip batches do not contain exactly five heads. A short record that misses perfect balance is not surprising under the fair model. The head-count standard deviation is √(10×0.5×0.5), about 1.5811 heads.
A past imbalance does not create a debt
After several tails, the next fair independent flip still has probability one-half of heads. The model does not owe a compensating head. Over longer records, a proportion can move closer to one-half while the absolute difference between total heads and total tails grows. These are different ways to describe convergence, so “the counts must catch up” is a misleading interpretation.
Separate exploration from a test of fairness
Use the Coin Flipper to explore short runs, and record repeated batches using a plan set in advance. A formal assessment of a physical coin or random-number source requires a sampling protocol and a prespecified test; visual judgment of one sequence is not a certification. The head-count probability tool calculates model probabilities, while the streak tool addresses adjacent outcomes. The guide to recording simulation results explains how selective reporting can create apparent patterns that the original experiment did not establish.
Check the model behind your result
Use the Coin Flipper for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26