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Binomial Probability Calculator

Number of Trials (n)

Number of Successes (k)

Probability of Success (p)

0 1

Binomial Probability Calculator Guidelines

You’re just a few inputs away from powerful insights—let’s get started!

How to Use the Calculator

Step 1: Enter the number of trials (n). Must be a non-negative integer.
Step 2: Enter the number of successes (k). Must be between 0 and n.
Step 3: Enter the probability of success (p) as a decimal between 0 and 1.
Step 4: Click the Calculate button to display P(X = k).

Important Notes

Results are rounded to four decimal places.
The calculator automatically handles special cases (e.g. k > n returns 0).
Supports high values of n and advanced precision for small probabilities.
For cumulative probabilities (P(X ≤ k)), you must sum individual probabilities manually or use a dedicated cumulative tool.

Binomial Probability Calculator Description

What is Binomial Probability?

The binomial distribution calculates the probability of obtaining exactly k successes in n independent Bernoulli trials. Each trial results in either success or failure.

Success probability (p): constant for every trial
Failure probability (q): equal to (1 - p)
Trials are independent: the result of one doesn't affect the others

Probability Formula

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

Where:

C(n, k) = n! / (k!(n-k)!) is the binomial coefficient
n: number of trials
k: number of successes
p: probability of success
q = 1 - p: probability of failure

Key Properties

The distribution is discrete
Supports only integer values of k from 0 to n
Mean: μ = n × p
Variance: σ² = n × p × (1 - p)
Standard Deviation: σ = √(n × p × q)

Pro Tip: Use the binomial distribution when outcomes are binary, such as pass/fail, yes/no, or defective/working scenarios.

Understanding Edge Cases

k > n: Not possible, so P(X = k) = 0
n = 0: Only outcome is 0 successes, so P(X = 0) = 1
p = 0: Only P(X = 0) = 1 (no successes possible)
p = 1: Only P(X = n) = 1 (always succeeds)
p = 0.5: Distribution is symmetric about k = n/2
k = 0: P(X = 0) = (1 - p)^n
k = n: P(X = n) = p^n

Real-World Mini Case Studies

1. Clinical Trial: Drug Effectiveness

A medication is tested on 15 patients. With a 70% success rate, what's the probability exactly 12 recover?

P(X = 12) = C(15, 12) × (0.7^12) × (0.3^3) ≈ 0.2311

2. Microchip Manufacturing: Defect Rate

A batch of 100 microchips has a known 2% defect rate. What’s the chance exactly 2 are defective?

P(X = 2) = C(100, 2) × (0.02^2) × (0.98^98) ≈ 0.2707

3. Email Campaign: Open Rate

Out of 200 emails sent with a 25% open rate, what's the probability that exactly 50 users open it?

P(X = 50) = C(200, 50) × (0.25^50) × (0.75^150) ≈ 0.0481

4. Product Quality: High Success Rate

In a 20-unit test, if the success rate is 95%, what's the chance 18 pass?

P(X = 18) = C(20, 18) × (0.95^18) × (0.05^2) ≈ 0.1871

Get instant clarity—try the calculator now and make every decision count!

Example Calculation

Example Probability Table

n	k	p	P(X = k)
10	5	0.5	0.2461
20	10	0.4	0.1171
15	12	0.7	0.2311
100	2	0.02	0.2707
200	50	0.25	0.0481
20	18	0.95	0.1871
0	0	0.3	1.0
5	6	0.4	0.0