Probability Calculator N Choose K

This probability calculator helps you determine the probability of selecting exactly k successes from n trials when each trial has the same probability of success. It's commonly used in statistics, quality control, and probability theory.

What is n choose k?

The term "n choose k" refers to the number of ways to choose k items from a set of n items without regard to the order of selection. This is also known as a combination, calculated using the binomial coefficient formula.

In probability terms, when you have n independent trials with the same probability of success (p) on each trial, the probability of getting exactly k successes is calculated using the binomial probability formula:

Binomial Probability Formula

P(X = k) = C(n, k) × p^k × (1-p)^n-k

Where:

C(n, k) = n choose k = number of combinations
p = probability of success on a single trial
n = total number of trials
k = number of successes

The combination C(n, k) is calculated as:

Combination Formula

C(n, k) = n! / (k! × (n - k)!)

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

This calculator uses these formulas to compute the probability of exactly k successes in n trials.

Probability Formula

The complete probability formula for exactly k successes in n trials is:

Complete Probability Formula

P(X = k) = [n! / (k! × (n - k)!)] × p^k × (1-p)^n-k

This formula combines the combination calculation with the probability of success and failure on each trial.

Key Assumptions

Trials are independent
Each trial has the same probability of success (p)
Only two possible outcomes for each trial (success/failure)

How to Use This Calculator

Enter the total number of trials (n)
Enter the number of successes (k)
Enter the probability of success on a single trial (p)
Click "Calculate" to see the probability
Review the result and chart visualization

The calculator will display the probability of exactly k successes in n trials, along with a visual representation of the probability distribution.

Examples

Example 1: Coin Flips

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

Using the calculator:

n = 10
k = 6
p = 0.5

The calculator would show a probability of approximately 0.2051 or 20.51%.

Example 2: Quality Control

A factory produces light bulbs with a 95% success rate. If 20 bulbs are tested, what's the probability that exactly 19 work?

Using the calculator:

n = 20
k = 19
p = 0.95

The calculator would show a probability of approximately 0.0950 or 9.50%.

FAQ

What is the difference between n choose k and permutations?: n choose k (combinations) counts the number of ways to choose k items without regard to order, while permutations count the number of ways to arrange k items in order.
When should I use this calculator?: Use this calculator when you need to find the probability of exactly k successes in n independent trials with the same probability of success.
What if my probability of success is not between 0 and 1?: The calculator will show an error if you enter a probability outside the 0 to 1 range. Please enter a valid probability between 0 and 1.
Can I use this calculator for continuous variables?: No, this calculator is designed for discrete trials with binary outcomes (success/failure). For continuous variables, you would need a different probability distribution.
How accurate are the results?: The calculator uses standard binomial probability formulas and provides results with up to 4 decimal places of precision.