Why isn't the probability of 5 heads in 10 flips exactly 50%?

Getting exactly 5 heads is just one of many possible outcomes (0 through 10 heads); the binomial formula C(n,k) × p^k × (1−p)^(n−k) spreads probability across all outcomes, so any single exact count, even the most likely one, is well under 50%.

Does this calculate "exactly k" or "at least k"?

The primary result is the probability of exactly the target count. "At least" probabilities (summing the binomial probability across all counts from the target upward) are also shown as a separate, related output.

Coin Flip & Dice Probability Calculator

Calculate the probability of a given number of heads or matching dice rolls.

How this is calculated

P(exactly k successes in n trials) = C(n,k) × p^k × (1−p)^(n−k)

Assumptions used in this calculation

Fair coin / fair die: Assumes a fair coin (p=0.5 per flip) or a fair, unweighted die (p=1/6 per face), with each trial independent of the others.

About this calculator

The coin flip and dice probability calculator uses the binomial distribution to find the probability of exactly (or at least) a target number of heads across coin flips, or matching faces across dice rolls.

Worked example

Exactly 5 heads in 10 flips

Result: 24.61%

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Frequently asked questions

Probability Calculator

Calculate the probability of two independent events happening together, either, or neither.

Coin Flip & Dice Probability Calculator

About this calculator

Frequently asked questions

Why isn't the probability of 5 heads in 10 flips exactly 50%?

Does this calculate "exactly k" or "at least k"?

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