Binomial Distribution Calculator

Calculate binomial probabilities, cumulative probabilities, expected value, and variance. Interactive distribution chart with step-by-step solutions.

Formula:P(X=k) = C(n,k) x p^k x (1-p)^(n-k)

Results

P(X = k)

24.6094%

Probability of exactly 5 successes

P(X <= 5)62.3047%
P(X >= 5)62.3047%
Expected Value5.0000
Std Deviation1.5811

Parameters

Total number of independent trials

Number of successes to calculate probability for

0.5
01

Probability of success on each trial

Practical Examples

Probability Results

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

Probability Distribution

Green bar shows P(X = 5). Red dashed line shows expected value E[X] = 5.00.

Step-by-Step Calculation

Step 1:P(X = 5) = C(10, 5) x 0.5^5 x 0.5000^5
Step 2:C(10, 5) = 10! / (5! x 5!) = 252
Step 3:0.5^5 = 3.125000e-2
Step 4:0.5000^5 = 3.125000e-2
Step 5:P(X = 5) = 252 x 3.1250e-2 x 3.1250e-2
Step 6:P(X = 5) = 0.24609375

Probability Table

kP(X = k)P(X <= k)P(X >= k)
00.0977%0.0977%100.0000%
10.9766%1.0742%99.9023%
24.3945%5.4688%98.9258%
311.7188%17.1875%94.5313%
420.5078%37.6953%82.8125%
524.6094%62.3047%62.3047%
620.5078%82.8125%37.6953%
711.7188%94.5313%17.1875%
84.3945%98.9258%5.4688%
90.9766%99.9023%1.0742%
100.0977%100.0000%0.0977%

Distribution Statistics

Expected Value

E[X] = n x p

E[X] = 10 x 0.5

5.0000

Variance

Var = n x p x (1-p)

Var = 10 x 0.5 x 0.5000

2.5000

Standard Deviation

SD = sqrt(Variance)

SD = sqrt(2.5000)

1.5811

Normal Approximation

Conditions: np >= 10 and n(1-p) >= 10

np = 5.00 (not valid)n(1-p) = 5.00 (not valid)

Binomial Distribution Formulas

Probability Mass Function

P(X=k) = C(n,k) x p^k x (1-p)^(n-k)

Binomial Coefficient

C(n,k) = n! / (k! x (n-k)!)

Expected Value

E[X] = n x p

Variance

Var(X) = n x p x (1-p)

Results

P(X = k)

24.6094%

Exactly 5 successes

P(X <= 5)62.3047%
Expected Value5.00

?What is Binomial Distribution?

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.

What is Binomial Distribution?

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.

Key Facts About Binomial Distribution

  • P(X=k) = C(n,k) x p^k x (1-p)^(n-k)
  • Expected value (mean): E[X] = n x p
  • Variance: Var(X) = n x p x (1-p)
  • Standard deviation: sigma = sqrt(n x p x (1-p))
  • Requires n independent trials with constant probability p
  • Each trial has exactly two outcomes: success or failure
  • P(X <= k) is cumulative probability up to k successes
  • Normal approximation valid when n x p >= 10 and n x (1-p) >= 10

Quick Answer

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.

Frequently Asked Questions

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