How does the coin flip simulator work?

Each flip draws a random number between 0 and 1. If it is less than p (default 0.5), the flip is counted as heads; otherwise tails. We use JavaScript's Math.random(), which is a high-quality pseudo-random generator suitable for probability demonstrations. For cryptographic randomness you would want crypto.getRandomValues(), but Math.random() is statistically indistinguishable from fair for this use case.

What is the probability of getting exactly N heads in M flips?

Use the binomial formula P(X = N) = C(M, N) × p^N × (1-p)^(M-N), where C(M, N) is the binomial coefficient and p is the probability of heads (0.5 for a fair coin). Our Probability mode computes this exactly for up to 1,000 flips. Example: exactly 7 heads in 10 flips at p = 0.5 is C(10,7) × 0.5^10 = 120/1024 ≈ 11.72%.

What is the probability of at least 60 heads in 100 flips?

For a fair coin: about 2.8%. The expected number of heads is 50, with standard deviation √(100 × 0.5 × 0.5) = 5. Getting 60+ is 2 standard deviations above the mean. Our Probability mode sums P(X = 60) + P(X = 61) + ... + P(X = 100) for the exact answer.

Can I simulate a biased (unfair) coin?

Yes — the bias slider lets you set the heads probability anywhere from 0 (always tails) to 1 (always heads). A value of 0.6 simulates a coin that lands heads 60% of the time on average. Real-world biased coins from physical defects rarely exceed a 0.51 bias, but for teaching or what-if scenarios the slider accepts any value in [0, 1].

Is a real coin actually 50/50?

Very close. Persi Diaconis and colleagues showed that a spinning coin has a tiny bias toward landing on the side facing up when flipped (around 0.51, not 0.50), but this requires physical study — the digit-on-the-side effect is negligible. For most practical purposes, treating a flipped coin as 50/50 is accurate to within 1%.

Coin Flip Calculator

The Coin Flip Calculator runs two modes. In Simulator mode you flip a virtual coin anywhere from 1 to 10,000 times and watch the heads/tails counts and percentages drift toward the expected 50/50 split (or your chosen bias). In Probability mode it computes the exact binomial probability of getting exactly N heads, at most N, or at least N heads in M flips using the formula P(X = k) = C(n,k) × p^k × (1-p)^(n-k). A bias slider lets you simulate loaded coins — set heads probability anywhere from 0 to 1. Use it for probability homework, decision-making, teaching the law of large numbers, or just settling a dispute.

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calendar_today Updated: September 2025

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Coin Flip

Heads Probability (bias) 0.50

Always tails (0) Fair (0.5) Always heads (1)

Number of Flips

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Heads

0 0%

Tails

0 0%

Total Flips

Expected Heads

Actual vs Expected

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Longest Streak

history

Recent Flips (last 100)

Flip the coin to start.

tips_and_updates Tips

• A fair coin has p = 0.5 — set the bias slider to 0.5 for an unbiased simulator
• Real coins are very close to fair; studies measure bias under 0.51
• The 'law of large numbers' says results approach the true probability as flips grow — 10 flips can easily deviate 30% from 50/50, but 10,000 flips rarely deviate more than 1%
• Getting exactly 50% heads is less likely than you think — with 10 flips it is only 24.6%
• Long streaks are normal: in 100 flips of a fair coin, expect a run of 6+ heads or tails roughly 80% of the time
• Use Probability mode for homework — it gives exact binomial answers, no sampling noise
• A bias of 0.6 (slightly loaded coin) flips heads 60% of the time on average

How to Use the Coin Flip

Pick a mode

Simulator flips a virtual coin up to 10,000 times. Probability computes exact binomial probabilities.

Set number of flips

In Simulator mode, this is how many times to flip. In Probability mode, this is M — the total trials.

Set target heads (Probability mode)

N = how many heads you are asking about. We show P(exactly N), P(at most N), and P(at least N).

Adjust bias if needed

Default 0.5 is a fair coin. Slide to 0.6 for a slightly loaded heads coin, 0.4 for tails-biased, etc.

The Formula

Each coin flip is an independent Bernoulli trial with probability p of heads. For n independent flips, the number of heads follows a binomial distribution. The probability of exactly k heads is C(n,k) × p^k × (1-p)^(n-k). 'At most k' and 'at least k' probabilities are cumulative sums. When p = 0.5 (fair coin), the distribution is symmetric around n/2 and the most likely outcome is exactly n/2 heads.

P(X = k) = C(n, k) × p^k × (1-p)^(n-k)

lightbulb Variables Explained

n Total number of flips
k Number of heads outcomes of interest
p Probability of heads on a single flip (default 0.5)
C(n,k) Binomial coefficient = n! / (k! × (n-k)!)

tips_and_updates Pro Tips

A fair coin has p = 0.5 — set the bias slider to 0.5 for an unbiased simulator

Real coins are very close to fair; studies measure bias under 0.51

The 'law of large numbers' says results approach the true probability as flips grow — 10 flips can easily deviate 30% from 50/50, but 10,000 flips rarely deviate more than 1%

Getting exactly 50% heads is less likely than you think — with 10 flips it is only 24.6%

Long streaks are normal: in 100 flips of a fair coin, expect a run of 6+ heads or tails roughly 80% of the time

Use Probability mode for homework — it gives exact binomial answers, no sampling noise

A bias of 0.6 (slightly loaded coin) flips heads 60% of the time on average

The coin flip is the simplest and most intuitive example of a random binary event, making it a cornerstone of probability theory and statistics education. A fair coin has exactly a 50% chance of landing heads and 50% chance of tails on any single flip, but the actual observed distribution in a small sample often deviates from this expectation. Flip a coin 10 times and getting 7 heads is not unusual — it happens about 11.7% of the time. Flip it 1,000 times and the proportion of heads almost certainly falls between 47% and 53%, illustrating the law of large numbers. Coin flips follow the binomial distribution, which gives the exact probability of observing k heads in n flips. This distribution underlies hypothesis testing, confidence intervals, and A/B testing in modern statistics. This coin flip calculator operates in two modes: a simulator that flips a virtual coin up to 10,000 times with animated results and running statistics, and a probability calculator that computes exact binomial probabilities for any number of trials and desired outcomes. Use it for classroom demonstrations, quick decisions, game night, or exploring how randomness behaves at different sample sizes.

Simulator mode vs probability mode

Simulator mode runs actual random coin flips and reports observed heads and tails counts plus a running percentage chart. Because it uses real randomness, 100 flips might give you 47 heads or 56 heads — you can hit 'Flip' again to see the variance. Probability mode computes exact binomial probabilities with no randomness: the answer is deterministic and matches textbook values. Use Simulator to teach the law of large numbers; use Probability for homework and exact answers.

Law of large numbers, streaks, and how to spot a biased coin

With 10 flips of a fair coin, deviation from 50/50 is huge — a 7-3 split happens 11.7% of the time. With 1,000 flips, a 7-3 style (700 heads) would be astronomically unlikely (P ≈ 10^-38). Streaks are common: in 100 fair flips, the longest run of heads averages about 7. To detect a biased coin, run at least 1,000 flips and compare to the expected standard deviation of √(n × p × (1-p)). Results more than 3 SDs from the mean strongly suggest bias.

Frequently Asked Questions

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Coin Flip Calculator

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Recent Flips (last 100)

tips_and_updates Tips

How to Use the Coin Flip

Pick a mode

Set number of flips

Set target heads (Probability mode)

Adjust bias if needed

The Formula

lightbulb Variables Explained

tips_and_updates Pro Tips

Simulator mode vs probability mode

Law of large numbers, streaks, and how to spot a biased coin

Frequently Asked Questions

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