Calculate binomial probabilities, cumulative probabilities, expected value, and variance. Interactive distribution chart with step-by-step solutions.
P(X = k)
24.6094%
Probability of exactly 5 successes
Total number of independent trials
Number of successes to calculate probability for
Probability of success on each trial
P(X = 5)
24.6094%
Exactly 5 successes
P(X <= 5)
62.3047%
At most 5 successes
P(X >= 5)
62.3047%
At least 5 successes
P(X < 5)
37.6953%
P(X > 5)
37.6953%
Expected Value
5.00
Std Deviation
1.5811
Green bar shows P(X = 5). Red dashed line shows expected value E[X] = 5.00.
| k | P(X = k) | P(X <= k) | P(X >= k) |
|---|---|---|---|
| 0 | 0.0977% | 0.0977% | 100.0000% |
| 1 | 0.9766% | 1.0742% | 99.9023% |
| 2 | 4.3945% | 5.4688% | 98.9258% |
| 3 | 11.7188% | 17.1875% | 94.5313% |
| 4 | 20.5078% | 37.6953% | 82.8125% |
| 5 | 24.6094% | 62.3047% | 62.3047% |
| 6 | 20.5078% | 82.8125% | 37.6953% |
| 7 | 11.7188% | 94.5313% | 17.1875% |
| 8 | 4.3945% | 98.9258% | 5.4688% |
| 9 | 0.9766% | 99.9023% | 1.0742% |
| 10 | 0.0977% | 100.0000% | 0.0977% |
E[X] = n x p
E[X] = 10 x 0.5
5.0000
Var = n x p x (1-p)
Var = 10 x 0.5 x 0.5000
2.5000
SD = sqrt(Variance)
SD = sqrt(2.5000)
1.5811
Conditions: np >= 10 and n(1-p) >= 10
P(X=k) = C(n,k) x p^k x (1-p)^(n-k)
C(n,k) = n! / (k! x (n-k)!)
E[X] = n x p
Var(X) = n x p x (1-p)
P(X = k)
24.6094%
Exactly 5 successes
The binomial distribution models the probability of k successes in n independent trials, each with probability p of success. Formula: P(X=k) = C(n,k) x p^k x (1-p)^(n-k). Expected value: E[X] = n x p. Variance: Var(X) = n x p x (1-p). Standard deviation: sigma = sqrt(n x p x (1-p)). Use for coin flips, quality control, surveys, and any repeated yes/no experiments.
The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. Each trial has exactly two outcomes (success/failure) with a constant probability of success. It is defined by two parameters: n (number of trials) and p (probability of success on each trial). Common applications include coin flipping, quality control testing, survey responses, and medical trials.
Quick-start with common scenarios
Test your skills with practice problems
Practice with 5 problems to test your understanding.
Explore similar calculators
The binomial distribution models the probability of k successes in n independent trials, each with probability p of success. Formula: P(X=k) = C(n,k) x p^k x (1-p)^(n-k). Expected value: E[X] = n x p. Variance: Var(X) = n x p x (1-p). Standard deviation: sigma = sqrt(n x p x (1-p)). Use for coin flips, quality control, surveys, and any repeated yes/no experiments.
A binomial distribution models the probability of getting exactly k successes in n independent trials, where each trial has probability p of success. It applies to situations with exactly two outcomes (yes/no, heads/tails, pass/fail).
P(X = k) = C(n,k) x p^k x (1-p)^(n-k), where C(n,k) is the binomial coefficient "n choose k" = n!/(k!(n-k)!). p^k is probability of k successes, (1-p)^(n-k) is probability of (n-k) failures.
Expected value E[X] = n x p. For example, if you flip a fair coin 10 times, the expected number of heads is 10 x 0.5 = 5. This is the long-run average number of successes.
Variance = n x p x (1-p). Standard deviation is the square root: SD = sqrt(n x p x (1-p)). For 10 fair coin flips: variance = 10 x 0.5 x 0.5 = 2.5, SD = 1.58.
The normal approximation is valid when both n x p >= 10 and n x (1-p) >= 10. Use mean = np and standard deviation = sqrt(np(1-p)). Apply continuity correction for better accuracy.
Last updated: 2025-01-15
P(X = k)
24.6094%
Probability of exactly 5 successes