# Coin Flip Probability Calculator

Calculate coin flip probabilities for fair or biased coins. Find the chance of getting exactly, at least, or at most a specific number of heads.

## What this calculates

Calculate the probability of getting a specific number of heads in a series of coin flips. Supports fair coins (50/50) and biased coins with any probability of heads.

## Inputs

- **Number of Flips** — min 1, max 170 — Total number of coin flips.
- **Target Heads Count** — min 0 — Desired number of heads.
- **Comparison** — options: Exactly, At Least, At Most — Whether you want exactly, at least, or at most the target count.
- **Coin Bias (P of Heads)** — min 0, max 1 — Probability of heads on each flip (0.5 for fair coin).

## Outputs

- **Probability** — The probability as a decimal.
- **Probability (%)** — The probability as a percentage.
- **Expected Heads** — Expected number of heads (n x p).
- **Standard Deviation** — Standard deviation of the number of heads.

## Details

Coin flip probability follows the binomial distribution, one of the most fundamental distributions in statistics. Each flip is an independent Bernoulli trial with two outcomes.

The Formula: P(X = k) = C(n, k) x p^k x (1-p)^(n-k), where n is the number of flips, k is the target number of heads, and p is the probability of heads on each flip. For a fair coin, p = 0.5.

Practical Applications: Coin flip probability is used in hypothesis testing (is this coin fair?), game theory, decision making, and as a foundation for understanding more complex probability distributions. The expected number of heads is n x p, and the standard deviation is the square root of n x p x (1-p).

## Frequently Asked Questions

**Q: What is the probability of getting exactly 5 heads in 10 flips?**

A: For a fair coin, the probability of exactly 5 heads in 10 flips is C(10,5) x 0.5^10 = 252/1024 = approximately 24.6%. This is the most likely single outcome, but there is still a 75.4% chance of getting some other number of heads.

**Q: How do you detect a biased coin?**

A: Flip the coin many times and compare the results to what a fair coin would produce. For example, if you flip 100 times and get 65 heads, you can use a binomial test: the probability of 65 or more heads with a fair coin is about 0.18%, suggesting the coin is biased. More flips give more statistical power.

**Q: What is the Gambler's Fallacy?**

A: The Gambler's Fallacy is the mistaken belief that past coin flip outcomes affect future ones. If you flip 5 heads in a row, the next flip still has a 50% chance of heads (for a fair coin). Each flip is independent, and the coin has no memory.

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Source: https://vastcalc.com/calculators/statistics/coin-flip
Category: Statistics
Last updated: 2026-04-21