Binomial Probability Distribution Calculator

The Binomial Probability Distribution Calculator finds the exact and cumulative probabilities. Input the number of trials, n, the probability of a successful trial, p, and the number of successful trials we wish to observe, k. The calculator outputs the exact and all cumulative probabilities.

Probability Calculator

Input the parameters for the binomial distribution, then click calculate to get exact and cumulative probabilities.

Binomial Distribution: X ~ Bin(n,p)

Number of trials (n):

Probability of success (p):

Number of successes (k):

Understanding the Binomial Distribution

The binomial distribution is a discrete probability distribution that is used to find the probability of a certain number of successes in a sequence of independent and identical trials. The probability of getting exactly k successes in n trials with a probability p is given by the following formula:

Binomial Probability Equation

$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$

Example Usage

John flips a fair coin 10 times.

a. What is the probability of getting exactly 6 heads?

b. What is the probability of getting at least 6 heads?

Given in the example, there are 10 flips so n = 10, the coin is fair sop = 0.5, and k = 6. Plugging this into the calculator, we get

Binomial Calculator Example

Number of trials (n):

Probability of success (p):

Number of successes (k):

Exact Probability:

P(X = 6) = 0.205078

Probability of exactly 6 successful trials is 20.5078%

Cumulative Probability:

P(X ≥ 6) = 0.376953

Probability of at least 6 successful trials is 37.6953%

P(X > 6) = 0.171875

Probability of more than 6 successful trials is 17.1875%

P(X ≤ 6) = 0.828125

Probability of at most 6 successful trials is 82.8125%

P(X < 6) = 0.623047

Probability of less than 6 successful trials is 62.3047%

Therefore, (a) the probability of getting exactly 6 heads is approximately 0.2051%, and (b) the probability of getting at least 6 heads is approximately 0.3770.