Binomial Probability Distribution Calculator

• Input parameters n (number of trials) and p (probability of success)
• Select “Exact and Cumulative” for k successes or “Between” for a range of successes
• Click “Calculate” to get the probabilities

For help using the calculator, see the example problems below. To learn more about the binomial distribution, read the introductory lesson on binomial random variables.

Probability Calculator

Binomial Distribution: X ~ Bin(n,p)

Number of trials (n):

Probability of success (p):

Calculation type: Exact and Cumulative Between

Number of successes (k):

Lower bound (k₁):

Upper bound (k₂):

Calculate Reset

Understanding the Binomial Distribution

The binomial distribution is a discrete probability distribution used to find the probability of getting 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 binomial probability formula.

Binomial Probability Formula

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

To use the binomial probability formula, the situation must meet the conditions of the binomial setting.

1. There is a fixed number of identical trials, denoted as n
2. Each trial results in one of two outcomes: success (S) or failure (F)
3. The probability of success is the same for every trial, denoted by p, where 0 $\leq$ p $\leq$ 1. The probability of failure is q = (1-p)
4. The trials are independent

The mean and standard deviation of a binomial distribution are given by the following formulas:

Binomial Mean and Standard Deviation Formulas

$\mu_Y = np$

$\sigma_Y = \sqrt{np(1-p)}$

Example Usage

1. 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?

   c. What is the mean and standard deviation of the number of heads?

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

Binomial Calculator "Exact & Cumulative" Example

X ~ Bin(10, 0.5), k = 6

Exact Probability:

P(X = 6) = 0.205078

Probability of exactly 6 successful trials is 20.5078%

Cumulative Probabilities:

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 fewer than 6 successful trials is 62.3047%

Distribution Statistics:

Mean (μ) = 5.0000

The average number of successes is about 5.00.

Variance (σ²) = 2.5000

The squared average deviation from the mean is 2.50.

Standard Deviation (σ) = 1.5811

The number of successes typically differs from the mean by about 1.58.

Therefore, (a) the probability of getting exactly 6 heads is approximately 0.2051, (b) the probability of getting at least 6 heads is approximately 0.3770, and (c) the mean is 5 with the standard deviation being approximately 1.5811.

We can also use the binomial calculator to find the probability of getting a certain amount of successes between two values. For example, if we want to find the probability of getting between 4 and 6 heads in 10 flips, we can use the “Between” option in the calculator.

2. John flips a fair coin 10 times.

   a. What is the probability of getting between 4 and 6 heads in 10 flips?

   We are given n = 10, p = 0.5, $k_1$ = 4, and $k_2$ = 6. Entering these values into the calculator, we get

   Binomial Calculator "Between" Example

   X ~ Bin(10, 0.5), k₁ = 4, k₂ = 6

   Between Probability:

   P(4 ≤ X ≤ 6) = 0.656250

   Probability of getting between 4 and 6 successful trials is 65.6250%

   Distribution Statistics:

   Mean (μ) = 5.0000

   The average number of successes is about 5.00.

   Variance (σ²) = 2.5000

   The squared average deviation from the mean is 2.50.

   Standard Deviation (σ) = 1.5811

   The number of successes typically differs from the mean by about 1.58.

   (a) The probability of getting between 4 and 6 heads in 10 flips is approximately 0.6563 or 65.63%.