Coin Flip Probability: A 10-Coin Classroom Experiment
A fair coin has a 50% chance of heads on each flip. That does not mean ten flips must produce five heads. Use this short experiment to show the difference between the probability of a single event and the results of a small sample.
Try ten flips
This simulator starts with ten coins. Generate a batch and record the heads and tails before generating again.
For a full-page display, open the 10-coin simulator. You can also flip one coin to build the sample one result at a time.
A ten-minute classroom activity
- Ask everyone to predict how many heads the first ten flips will contain. Record the predictions before drawing results.
- Flip ten coins. Write down the number of heads, the number of tails, and the fraction of heads. Seven heads out of ten is 7/10, or 70%.
- Repeat for ten batches, keeping all results. Add the heads from all batches and divide by 100. Do not discard batches that look unusual.
- Compare each ten-flip result with the combined hundred-flip result. Discuss how the larger sample changes the estimate.
- Repeat with 100 coins at once, or combine results across the class. A larger sample will usually have a proportion closer to 50%, although it is not guaranteed on any particular run.
Keep the raw counts alongside percentages. A result of 60% heads from ten flips contains much less evidence about the coin than 60% heads from ten thousand flips.
What results should you expect from ten fair flips?
Ten independent coin flips have 2¹⁰ = 1,024 equally likely sequences. For example, HHHHHTTTTT is one sequence; alternating heads and tails is another. Many different sequences contain five heads, while only one sequence contains ten heads.
| Outcome | Matching sequences | Probability |
|---|---|---|
| Exactly five heads | 252 | 24.61% |
| Exactly four or six heads | 420 | 41.02% |
| All heads | 1 | 0.0977% |
| All tails | 1 | 0.0977% |
| All ten the same, either heads or tails | 2 | 0.1953%, or 1 in 512 |
These values come from the binomial formula: the probability of exactly k heads in n fair flips is C(n, k) / 2ⁿ, where C(n, k) counts the ways to choose the positions of those heads. For five heads in ten flips, C(10, 5) = 252.
Is heads due after a run of tails?
No. If flips are independent and the coin is fair, the next flip still has a 50% chance of heads after five tails. Earlier outcomes do not change the next draw. The idea that the next result must compensate for a streak is called the gambler's fallacy.
A short streak also does not prove the generator is biased. To investigate fairness, choose your sample size and measurement before drawing, keep all observations, and compare the results with the variation expected under a fair model.
How the online coin works
Randomify's coin tool draws a random integer from 0 to 1 for each flip and maps it to heads or tails. It uses the browser's Web Crypto API when available, with a Math.random() fallback on browsers without that API. The tool's session history lets you inspect earlier batches. The calculations above assume independent, fair flips; a finite experiment will fluctuate around those expectations.
For another probability lesson, use the dice roller and compare how often each face appears. A coin has two possible outcomes, while a six-sided die has six.