Dice Roll Simulator
Roll virtual dice with 4, 6, 8, 10, 12, or 20 sides, combine multiple dice, and review totals, averages, and distributions.
Coin Flip Simulator
Flip a virtual coin one or multiple times and track heads-versus-tails counts and percentages for quick experiments or games.
1 to 10,000 per run
Repeat the experiment 1-100 times
Exact heads to compute probability for
Advanced Options
Type yes to show theoretical stats
Planning notes, formulas, and examples
About the Coin Flip Simulator
The Coin Flip Simulator lets you flip a virtual coin any number of times and track the results. See the count and percentage of heads versus tails.
Coin flipping is the simplest example of a Bernoulli trial — an experiment with exactly two equally likely outcomes. It is used in probability education, decision-making, sports (kick-off selection), and as a randomization device.
Flip 1 coin for a quick decision, or flip 1,000 to see the law of large numbers in action as the heads/tails ratio converges toward 50/50.
When This Page Helps
No physical coin needed. Flip any number of times and see statistical results to explore probability concepts or make quick decisions.
How to Use the Inputs
- Choose the number of flips (1 to 10,000).
- Click Flip to simulate.
- View the count of heads and tails.
- See the percentage breakdown.
- Flip multiple times to observe convergence to 50%.
Formula used
P(Heads) = 0.5, P(Tails) = 0.5 (fair coin)
For n flips, expected heads = n/2
Standard deviation = √(n × 0.5 × 0.5) = √(n)/2Example Calculation
Result: e.g. Heads: 53, Tails: 47
Flipping 100 times with a fair coin, you expect about 50 heads. Getting 53 is well within normal variation (±5 for 100 flips).
Tips & Best Practices
- A fair coin has exactly 50/50 probability.
- With more flips, the percentage gets closer to 50% (law of large numbers).
- For 1 flip, the standard deviation is 0.5; for 100 flips it's 5.
- Coin flips are independent: previous results don't affect the next flip.
- The gambler's fallacy is believing a streak of heads makes tails "due."
- Real coins have a very slight bias (about 51/49) due to weight distribution.
Coin Flipping in Decision-Making
Coin flips are used in sports (NFL overtime possession), in resolving ties, and in everyday decision-making. Some people use a coin flip not for its result but to notice which outcome they were hoping for.
The Mathematics of Coin Flipping
The probability of exactly k heads in n flips is C(n,k)×(0.5)^n. The cumulative distribution helps answer questions like "What is the probability of getting at least 60 heads in 100 flips?"
Randomness and Simulation
Coin flips are the building block of Monte Carlo simulations. Complex probabilistic systems can be modeled by combining many simple random experiments.
Sources & Methodology
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Frequently Asked Questions
A mathematically ideal coin is 50/50. Real coins have a slight bias (about 51% for the side facing up at the start), but the effect is negligible for practical purposes.
As the number of flips increases, the observed proportion of heads approaches the theoretical probability (50%). With 10 flips you might get 70% heads; with 10,000 it will be near 50%.
Yes. The probability is (0.5)^10 = 1/1024 ≈ 0.1%. Rare but not impossible. In 10,000 flips you would expect about 10 such streaks.
A Bernoulli trial is an experiment with exactly two outcomes (success/failure) with a fixed probability. Coin flipping is the classic example.
The number of heads in n fair coin flips follows a binomial distribution with p=0.5. Its mean is n/2 and standard deviation is √n/2.
The mistaken belief that past outcomes affect future independent events ("5 heads in a row — tails is due"). Each flip is independent with P(H)=50% regardless of history.
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