Binomial Distribution Calculator
Calculate exact and cumulative binomial probabilities for n trials. Get P(X = k), P(X ≤ k) and P(X ≥ k) with mean and variance.
P(X = k)
24.6094%
P(X ≤ k)
62.3047%
P(X ≥ k)
62.3047%
Mean (np)
5
Variance
2.5
Std. deviation
1.5811
Each trial must be independent with the same success probability, e.g. coin flips or defective-item checks.
Last updated: August 2026
How this calculator is verified
Checked by True Calculator automated test suite on
- Formula verified against a published worked example in the automated test suite
- Edge cases (zero, negative, boundary and unit-mismatch inputs) covered by unit tests
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When to Use This Calculator
Use the binomial distribution calculator for any repeated independent trial with two outcomes: counting heads in coin flips, defective items in a production batch, yes responses in a survey, or sixes in a set of cricket deliveries. Indian quality-control teams use it to decide whether a shipment's defect count is within acceptable limits, and students use it to solve probability problems from board exams and JEE papers. Betting and board-game players apply it to situations like rolling a specific number on a die — each roll is a trial with success probability 1/6. Because the calculator reports exact, cumulative and at-least probabilities together, it covers the three question styles that appear in exams and real decision-making, with the mean np and variance np(1−p) given for completeness.
How to Use This Calculator
- Step 1: Enter the number of trials n, such as 10 coin flips or 20 manufactured items checked.
- Step 2: Enter the number of successes k you want the probability for.
- Step 3: Enter the per-trial success probability as a percentage, for example 50 for a fair coin.
- Step 4: Read P(X = k), P(X ≤ k), P(X ≥ k) plus the mean, variance and standard deviation.
Worked Example
A dealer in Chandni Chowk tests a batch of 8 LED bulbs and wants the chance that exactly 3 are defective when the defect rate is 30%. Enter n = 8, k = 3 and probability 30%. The calculator returns P(X = 3) = 0.2541, about 25.4%, and P(X ≤ 3) = 0.8059, so there is an 80.6% chance of finding 3 or fewer defective bulbs. With 10 fair coin flips, exactly 5 heads has probability 0.2461 while 5 heads or fewer has probability 0.6230.
Tips and Common Mistakes
- •Tip 1: Use P(X ≥ k) for questions like 'what is the chance of at least 7 working bulbs' — it is 1 − P(X ≤ 6).
- •Tip 2: For 50:50 situations like coin flips, P(X ≥ k) equals P(X ≤ k) only when k is the midpoint — the calculator handles the symmetry for you.
- •Tip 3: Keep trials independent — drawing cards without replacement is not binomial because the probability shifts each draw.
- ✗Mistake 1: Entering the success probability as a fraction (0.5) when the field expects a percentage (50).
- ✗Mistake 2: Asking for k greater than n — the calculator returns an error hint, since more successes than trials is impossible.
Frequently Asked Questions
What is the binomial distribution?
It models the number of successes in a fixed number of independent trials where each trial has the same probability of success — like counting heads in 10 coin flips, or defective items in a batch. The distribution is defined by n (number of trials) and p (success probability per trial), and this calculator also reports the mean np and variance np(1−p).
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the probability of exactly k successes, while P(X ≤ k) is the probability of k or fewer successes. For example, in 10 coin flips the chance of exactly 5 heads is 24.6%, but the chance of 5 heads or fewer is 62.3%. The calculator shows both, plus P(X ≥ k) which is 1 − P(X ≤ k−1).
Can I use it for dice rolls or survey responses?
Yes. Any repeated independent trial with two outcomes fits the binomial model. Rolling a specific face on a die has p = 1/6, a yes/no survey response has p equal to the expected yes rate, and a quality check counting defects uses p as the defect rate. Just set n as the number of repetitions and k as the count you are interested in.
What happens if the probability is expressed as a percentage?
Enter the probability as a whole percentage — for example 50 for a fair coin or 30 for a 30% defect rate. The calculator converts it to a fraction automatically. If you enter a probability outside 0–100%, it is clamped to the nearest valid value so the results always stay meaningful.
When should I use the normal approximation instead?
For large n (roughly above 20 with p near 0.5), the binomial distribution looks almost identical to a normal curve, so the normal distribution calculator gives a close match. This calculator sums exact probabilities, so it stays exact for n up to about 2000; beyond that it automatically falls back to a normal approximation with continuity correction.
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