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Statistics • Coin Toss & Binomial Probability Calculator

Coin Toss & Binomial Probability Calculator

Calculate the exact and cumulative binomial probability of getting a target number of heads across a set number of fair coin tosses.

Coin Toss & Binomial Experiment Odds

Calculate exact and cumulative probabilities of flipping k heads in n fair coin tosses.

Exact P(X = 5)
24.61%
Cumulative P(X ≥ 5):62.30%
Total Possible Sequences:2^10 = 1,024
P(k) = C(n,k) · pᵏ · (1-p)ⁿ⁻ᵏ
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Calculation Guide & Reference

Coin Toss Binomial Probability Mathematics

Coin toss calculator computes the exact and cumulative binomial probability of getting a target number of heads across a set number of fair coin tosses.

Standardized Mathematical Formula
P(X=k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ

The binomial probability formula gives the chance of exactly k successes (heads) in n independent trials (tosses), each with success probability p. For a fair coin, p = 0.5.

Variables:
n:Total number of coin tosses
k:Target number of heads
p:Probability of heads on a single toss (0.5 for a fair coin)
C(n,k):Binomial coefficient — number of ways to arrange k heads among n tosses
How It Works (Step-by-Step)
  • 1Enter the total number of coin tosses (n) and the target number of heads (k).
  • 2View the exact probability P(X = k), the cumulative probability P(X ≥ k), and the total number of possible outcome sequences (2ⁿ).
Real-World Numerical Example
Exactly 5 Heads in 10 Coin Tosses

Flip a fair coin 10 times — what is the probability of getting exactly 5 heads?

•C(10,5) = 252.
•P(X=5) = 252 × 0.5⁵ × 0.5⁵ = 252 × 0.5¹⁰ = 252 ÷ 1024 ≈ 24.61%.
•Total possible sequences: 2¹⁰ = 1,024.
Result: There is approximately a 24.61% chance of flipping exactly 5 heads in 10 tosses.
Calculation Best Practices & Tips
In medical screening and classification, prior prevalence dramatically affects posterior positive predictive values (Bayes’ Theorem).
Use Bessel’s correction (n - 1) when calculating sample standard deviation to avoid systematically underestimating population variance.
Check assumptions of normality and homoscedasticity before making inferences with linear regression and ANOVA.

Frequently Asked Questions (FAQ)

Because there are 1,024 total possible sequences of 10 tosses, and only 252 of them contain exactly 5 heads — 50% is the most likely single outcome, but it still competes against many other possible head-counts (4 heads, 6 heads, etc.).

Exact probability P(X=k) is the chance of precisely k heads; cumulative probability P(X≥k) sums the exact probabilities for k, k+1, ..., n heads — useful for questions like "at least 5 heads."

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