Binomial Distribution Calculator

What the calculator does

The calculator works with a situation that repeats the same yes-or-no experiment many times: flipping a coin, rolling a die and asking whether a six came up, inspecting parts and asking whether each is defective. In Probability mode it tells you how likely a given number of successes is. In Trials required mode it works backwards and tells you how many attempts you need to be confident of seeing a result.

In Probability mode five numbers are reported, each both as a decimal and as a percentage: the chance of exactly x events, of at most x (X ≤ x), of more than x (X > x), of fewer than x (X < x), and of at least x (X ≥ x).

How to use it

1. Enter the probability of success in a single trial. Type a decimal such as 0.3, or a fraction such as 1/6 for one face of a die.
2. In Probability mode, enter the number of trials n and the number of events x. Both must be whole numbers, and x cannot exceed n.
3. In Trials required mode, enter x and the desired probability as a decimal below 1, for example 0.95. The number of trials is not needed because it is what the tool solves for.
4. Press Calculate.

The single-trial probability must be strictly between 0 and 1. A probability of 0 or 1 leaves nothing random to calculate.

When a binomial distribution applies

A count X follows a binomial distribution when all of these hold:

• there is a fixed number n of trials;
• each trial has only two outcomes, called success and failure;
• the chance of success p is the same on every trial;
• trials are independent, so one outcome does not influence the next.

The expected number of successes is n × p and the spread is measured by the variance n × p × (1 − p). For 20 fair coin flips that is a mean of 10 heads and a standard deviation of about 2.24.

If you draw items from a small batch without putting them back, the trials are not independent and a hypergeometric model is more appropriate than the binomial. The combination calculator helps with the counting involved.

The formulas

P(X = x) = C(n, x) · p^x · (1 − p)^{n − x}C(n, x) = n! ÷ (x! (n − x)!) is the number of ways to choose which x trials succeed. See the combination calculator.

P(X ≤ x) = P(X = 0) + P(X = 1) + … + P(X = x)

The other three outputs follow from these two. Because the probabilities of all outcomes add up to one:

• P(X < x) = P(X ≤ x) − P(X = x)
• P(X > x) = 1 − P(X ≤ x)
• P(X ≥ x) = 1 − P(X < x)

A common slip is to mix up "at least" (X ≥ x, includes x) with "more than" (X > x, excludes it). They differ by exactly P(X = x).

Worked examples

Coin flips. You flip a fair coin 20 times and want 8 heads. With p = 0.5, n = 20, x = 8 the calculator shows:

Dice. Roll a die 10 times. With p = 1/6, the chance of exactly two sixes is 29.07%, of at most two sixes 77.52%, and of at least two sixes 51.55%. The chance of no six at all is (5/6)¹⁰, or 16.15%, so at least one six has probability 83.85%.

Quality control. A process makes 2% defective parts and you inspect 50. The expected number of defects is 50 × 0.02 = 1. The chance of exactly 3 defects is 6.07%, of 3 or fewer 98.22%, and of more than 3 only 1.78%. The chance of at least one defect is 1 − 0.98⁵⁰ = 63.58%.

A complete distribution: 10 fair coin flips

Trials required mode

Sometimes the question is reversed: how many attempts do I need before I am likely to see a result? In this mode the calculator finds the smallest n for which the chance of at least x events is greater than the probability you asked for.

For x = 1 there is a closed form. The chance of at least one success in n trials is 1 − (1 − p)ⁿ, so you need n ≥ ln(1 − c) ÷ ln(1 − p), where c is the desired probability.

n ≥ ln(1 − 0.95) ÷ ln(5/6) = 16.43 → 17 rollsRolls of a die needed to be 95% sure of seeing at least one six.

Sixteen rolls give a 94.59% chance and seventeen give 95.49%, so the answer is 17. For rarer events the numbers grow quickly: with p = 0.001 you need 4,603 trials for a 99% chance of at least one success. For x above 1 the calculator searches directly. For example, to have a 90% chance of at least 5 successes when p = 0.4 you need 18 trials.

The desired probability must be above 0 and below 1, and x must be at least 1.

Precision and limits

• Results are rounded to 14 decimal places and may show fewer digits when the value has fewer significant digits, so 0.5 appears as 0.50 and 1 as 1.
• Very large n is handled, but floating-point arithmetic limits accuracy to roughly 12 to 14 significant digits; do not rely on the last digits for n in the thousands.
• Percentages are rounded to two decimals, so a tiny probability can show as 0% even though the decimal beside it is not zero.
• For large n with p not too close to 0 or 1 (a common rule of thumb is np and n(1 − p) both at least 10), the normal distribution calculator gives a quick approximation.

For rolling several dice and summing them, use the dice probability calculator; for odds notation see the odds calculator.