About the Binomial Distribution Calculator
The binomial distribution gives the probability of a certain number of successes in a fixed number of independent trials, when each trial has the same chance of success. How likely are exactly 3 heads in 10 coin tosses? What is the chance that at least 2 of 20 components are faulty? How many of 50 patients might respond to a treatment that works 60 percent of the time? All are binomial questions.
This binomial distribution calculator finds the probability of exactly k successes, plus the cumulative probabilities of at most k, fewer than k, at least k and more than k. It also gives the mean, standard deviation and most likely number of successes, and says whether a normal approximation would be reasonable. Calculations are done in log space, so even 100,000 trials are handled accurately.
How to Use the Binomial Distribution Calculator
Enter the number of trials, n.
Enter the probability of success on each trial, as a percentage.
Enter the number of successes, k, that you are interested in.
All five probabilities appear, with the mean and standard deviation.
The Formula
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)
C(n, k) = n! ÷ (k! × (n − k)!) the number of ways to place k successes
mean = n × p
standard deviation = √(n × p × (1 − p))Cumulative probabilities add up the individual terms: P(X ≤ k) is the sum of P(X = 0) through P(X = k).
Step-by-Step Example
Exactly 3 heads in 10 tosses of a fair coin.
C(10, 3) = 120
p^3 = 0.5^3 = 0.125 (1 − p)^7 = 0.5^7 = 0.0078125
P(X = 3) = 120 × 0.125 × 0.0078125 = 120 ÷ 1,024 = 0.1172The probability is 11.72%. For the cumulative figures:
P(X ≤ 3) = (1 + 10 + 45 + 120) ÷ 1,024 = 176 ÷ 1,024 = 17.19%
P(X ≥ 3) = 1 − (1 + 10 + 45) ÷ 1,024 = 1 − 0.0547 = 94.53%
mean = 10 × 0.5 = 5, SD = √2.5 = 1.58A Quality-Control Example
10% of items are faulty. What is the chance a sample of 20 contains none?
P(X = 0) = 0.9^20 = 0.1216 → 12.16%
P(X ≥ 1) = 87.84%
mean = 20 × 0.1 = 2 faulty itemsMost samples will contain at least one faulty item, and on average about two.
When the Binomial Model Applies
Four conditions must hold, sometimes remembered as BINS: Binary outcomes (success or failure), Independent trials, a fixed Number of trials, and the Same probability of success each time. Drawing cards without replacement breaks independence, because each draw changes what remains; when the sample is a small part of a large population, the binomial is still a good approximation. If the number of trials is not fixed — for example, tossing until the first head — a different distribution, the geometric, applies.
"At Least" and "At Most"
Most practical questions are cumulative. "What is the chance of at least 8 correct answers out of 10 by guessing?" needs P(X ≥ 8), the sum of P(8), P(9) and P(10). Be careful with the boundaries: "more than 3" means 4 or more, and "fewer than 3" means 0, 1 or 2. The calculator shows all four versions so you can pick the right one without adding terms by hand.
Is a Result Unusual?
The binomial distribution is often used to judge whether an outcome could plausibly be luck. Suppose a student scores 8 or more out of 10 on a true-or-false quiz. By pure guessing, each answer has a 50 percent chance, and P(X ≥ 8) = 56 ÷ 1,024, about 5.47 percent — unlikely, but it happens to roughly one guesser in 18. On a multiple-choice quiz with four options per question, guessing succeeds only 25 percent of the time, and the chance of 8 or more correct falls to about 0.042 percent, roughly 1 in 2,400. A score like that almost certainly reflects real knowledge. This reasoning is the basis of the binomial test, which compares an observed count of successes with the count expected under a stated probability.
The Normal Approximation
For large n, the binomial distribution looks like a normal curve with mean np and standard deviation √(np(1 − p)). A common rule is that np and n(1 − p) should both be at least 10. The approximation was essential before computers, and it remains useful for quick mental estimates: in 400 tosses of a fair coin, the mean is 200 and the SD is 10, so about 95 percent of the time the number of heads is between about 180 and
- For smaller or more lopsided cases, exact binomial figures like these are more
accurate.
Understanding Your Result
The headline is the probability of exactly k successes.
The at most, fewer than, at least and more than lines give the cumulative probabilities.
The mean and SD line gives the expected number of successes, the spread and the most likely count.
The worth knowing line says whether a normal approximation would be reasonable.
When Should You Use This Calculator?
Use it for coin, dice and card problems with a fixed number of tries.
Use it for quality control and acceptance sampling.
Use it to judge whether a number of successes is unusual for a given success rate.
Use it for statistics homework on discrete distributions.
Common Mistakes
Using it when trials are not independent. Sampling without replacement from a small group needs the hypergeometric distribution.
Mixing up "at least" and "more than". At least 3 includes 3; more than 3 does not.
Entering the probability of failure. Enter the chance of the outcome you are counting.
Forgetting the C(n, k) term. There are many orders in which k successes can occur.
Applying the normal approximation to small samples. Use exact probabilities when np or n(1 − p) is below 10.
Rounding the probability of success too early. Small changes in p shift the tail probabilities noticeably, so keep full precision.