Coin Flip
Flip a virtual coin.
Input
Result
What is a Coin Flip Calculator?
A Coin Flip simulator replicates the randomness of a physical coin toss using cryptographic random number generation. It is used for making decisions, settling disputes, teaching probability concepts, or running statistical experiments. The virtual coin produces either heads or tails with a perfect 50/50 probability on each independent flip.
How to calculate coin flip probability (the formula)
Each flip is an independent Bernoulli trial with P(heads) = 0.5 and P(tails) = 0.5. The probability of getting exactly k heads in n flips follows the binomial distribution: P(X = k) = C(n,k) × 0.5^n where C(n,k) = n! / (k! × (n-k)!). The expected value E[X] = n × 0.5 = n/2 heads. The standard deviation is √(n × 0.5 × 0.5) = √n / 2.
Example calculation
Flip a coin 10 times. The probability of exactly 5 heads: C(10,5) × 0.5^10 = 252 × 1/1024 ≈ 24.6%. The probability of at least 1 head: 1 - P(all tails) = 1 - 0.5^10 = 1 - 1/1024 ≈ 99.9%. For 100 flips, the expected number of heads is 50, and 95% of experiments will yield between 40 and 60 heads (mean ± 2 standard deviations).
Common mistakes
- The gambler's fallacy — After five tails in a row, the next flip still has exactly 50% chance of heads. Past outcomes do not affect future independent flips.
- Assuming more flips guarantee exact 50/50 — While the proportion converges to 50%, the absolute difference between heads and tails grows as √n. More flips mean larger potential absolute deviations.
- Treating a single flip as meaningful — One flip has high variance. Reliable conclusions about fairness require hundreds of flips.
Frequently asked questions
What is the probability of heads or tails?
A fair coin has exactly 50% probability for heads and 50% for tails.
Is a virtual coin flip truly random?
Yes, it uses cryptographic random number generation equivalent to a physical coin toss.
Can I flip multiple coins at once?
Yes, you can specify the number of flips and see both individual results and totals.
What is the law of large numbers with coin flips?
As the number of flips increases, the ratio of heads to tails approaches 50/50.
What is a streak in coin flips?
A streak is consecutive identical outcomes, like five heads in a row.