Calculate the probability of getting a specific number of heads or tails in coin flips — exact, cumulative, streaks, and distribution tables.
The coin flip probability calculator computes exact probabilities for any number of coin flips. Whether you have a fair coin (p = 0.5) or a biased one, this tool calculates the probability of getting exactly k heads, at most k, at least k, or more than k — along with streak probabilities and the full distribution table.
Coin flipping is the simplest example of a Bernoulli process and follows the binomial distribution. For n flips with probability p of heads on each flip, the probability of exactly k heads involves the binomial coefficient C(n,k) multiplied by p^k × (1−p)^(n−k).
Beyond individual probabilities, the calculator estimates the chance of getting a streak of consecutive heads and computes expected statistical properties. Preset scenarios let you quickly explore common situations from 10-flip experiments to 1000-flip simulations. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
Coin flip probability is the gateway to understanding all of probability theory. It introduces binomial distributions, expected values, independence, and the law of large numbers in the most intuitive way possible.
This calculator is ideal for probability class exercises, settling bets, understanding gambling odds, and building intuition about random processes.
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). Expected heads: μ = np. Standard deviation: σ = √(np(1 − p)). Streak of s heads in n flips: P ≈ 1 − (1 − p^s)^(n/s).
Result: P(X = 5) ≈ 0.2461 (24.61%)
With 10 fair coin flips, the probability of exactly 5 heads is C(10,5) × 0.5^10 = 252/1024 ≈ 24.61%. The most likely outcome, but it happens less than 1 in 4 times.
As the number of flips grows, the proportion of heads converges to p. With 10 flips, getting 30% or 70% heads is common. With 10,000 flips, the proportion will almost certainly be between 49% and 51%. This convergence is the law of large numbers in action.
People vastly underestimate how common streaks are. In 100 fair coin flips, the expected longest streak of heads is about 7. Casinos and sports commentators frequently misinterpret natural streaks as evidence of "hot hands" or "cold streaks" when they're perfectly consistent with randomness.
While coin flips model fair experiments, biased coins (p ≠ 0.5) model many real scenarios: conversion rates, success probabilities, and binary outcomes where one result is more likely than the other. The binomial framework handles all of these identically.
Exactly 50% or 0.5. Each flip is independent — previous results don't influence future ones.
Research suggests real coins have a slight bias (about 51%) toward the side facing up before the flip. For practical purposes, this is negligible.
The mistaken belief that past results affect future independent events. After 5 heads in a row, the next flip is still 50/50 — the coin has no memory.
(1/2)^10 = 1/1024 ≈ 0.098%. Rare for a single attempt, but in 1000 sets of 10 flips, you'd expect about one such streak.
The calculator handles up to 1000 flips using exact binomial probabilities. For larger numbers, the normal approximation (μ = np, σ = √(np(1−p))) works well.
Flipping n coins simultaneously is mathematically identical to flipping one coin n times — each flip is independent with the same probability. Use this as a practical reminder before finalizing the result.