Binomial Probability Calculator N P X

This binomial probability calculator helps you determine the probability of exactly x successes in n independent trials, each with success probability p. The calculator uses the binomial probability formula to provide accurate results.

What is Binomial Probability?

Binomial probability refers to the probability of having exactly x successes in n independent trials, each with success probability p. It's commonly used in statistics and probability theory to model scenarios with two possible outcomes (success/failure).

The binomial distribution is characterized by the following properties:

Fixed number of trials (n)
Independent trials
Constant probability of success (p) for each trial
Only two possible outcomes: success or failure

Binomial probability is widely used in quality control, medical testing, sports analytics, and many other fields where binary outcomes are common.

Binomial Probability Formula

The probability of exactly x successes in n trials is given by:

P(X = x) = C(n, x) × p^x × (1-p)^n-x

Where:

C(n, x) is the combination of n items taken x at a time
p is the probability of success on a single trial
n is the number of trials
x is the number of successes

The combination C(n, x) can be calculated using the formula:

C(n, x) = n! / (x! × (n-x)!)

Where "!" denotes factorial, the product of all positive integers up to that number.

How to Use the Calculator

Enter the number of trials (n) in the first input field
Enter the probability of success (p) in the second input field (between 0 and 1)
Enter the number of successes (x) in the third input field
Click the "Calculate" button to compute the binomial probability
View the result and interpretation
Optionally, reset the calculator to start over

Note: The calculator will show an error if x is greater than n, or if p is outside the 0-1 range.

Binomial Probability Examples

Example 1: Coin Toss

If you toss a fair coin (p = 0.5) 10 times (n = 10), what's the probability of getting exactly 6 heads (x = 6)?

Using the formula:

P(X = 6) = C(10, 6) × (0.5)⁶ × (0.5)⁴ = 210 × 0.015625 × 0.0625 ≈ 0.2637 or 26.37%

Example 2: Quality Control

A factory produces light bulbs with a 95% success rate (p = 0.95). A quality inspector checks 20 bulbs (n = 20). What's the probability that exactly 19 bulbs are good (x = 19)?

Using the formula:

P(X = 19) = C(20, 19) × (0.95)¹⁹ × (0.05)¹ ≈ 20 × 0.5766 × 0.05 ≈ 0.5766 or 57.66%

Binomial Probability Table

The following table shows binomial probabilities for different values of n, p, and x:

n	p	x	Probability
5	0.5	2	0.3125
10	0.5	6	0.2637
20	0.95	19	0.5766
30	0.8	24	0.1259
50	0.7	35	0.1201

FAQ

What is the difference between binomial and normal distribution?: The binomial distribution models the number of successes in a fixed number of independent trials, while the normal distribution models continuous data that clusters around a mean.
When should I use binomial probability?: Use binomial probability when you have a fixed number of independent trials with two possible outcomes (success/failure) and a constant probability of success.
What happens if p is greater than 1?: The calculator will show an error message since probability values must be between 0 and 1.
Can I calculate cumulative binomial probability with this calculator?: No, this calculator computes exact binomial probabilities. For cumulative probabilities, you would need to sum probabilities for all x values up to your desired number of successes.
What's the difference between binomial and Bernoulli distribution?: The Bernoulli distribution is a special case of the binomial distribution where n=1 (single trial). The binomial distribution extends this to multiple trials.