Binomial Probability Calculator
Compute exact P(X=k) and cumulative binomial probabilities P(X≤k), P(X≥k), distribution mean, variance, and standard deviation.
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What Is a Binomial Probability Calculator?
The Binomial Probability Calculator evaluates the probability of achieving exactly or cumulatively k successes in n independent Bernoulli trials, where each trial has the same probability of success p.
How to Use This Calculator
- Enter the total number of independent trials (n).
- Enter the probability of success on any individual trial (p, between 0 and 1).
- Enter the target number of successes (k).
- Analyze exact probability P(X = k), cumulative bounds P(X ≤ k) and P(X ≥ k), and summary statistics (mean and variance).
Binomial PMF and Distribution Parameters
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad \mu = np, \quad \sigma^2 = np(1-p)Where n is number of trials, k is number of successes, p is success probability per trial, and (1-p) is failure probability.
Worked Example
Scenario: A basketball player with an 80% free throw percentage (p = 0.8) takes 10 shots. Find the probability of making exactly 8 shots.
Identify parameters: n = 10, p = 0.8, k = 8.
Compute combinations: C(10, 8) = 45.
Calculate powers: 0.8^8 ≈ 0.16777, and (1 - 0.8)^2 = 0.2^2 = 0.04.
Multiply: 45 × 0.16777 × 0.04 = 0.30199 (30.20%).
Tips & Key Notes
- A binomial experiment requires four conditions (BINS): Binary outcomes (success/failure), Independent trials, Fixed Number of trials n, and Same probability p on each trial.
- When n is large and p is close to 0.5 (np ≥ 10 and n(1-p) ≥ 10), the binomial distribution can be closely approximated by a normal distribution.
Frequently Asked Questions
What are Bernoulli trials?
A Bernoulli trial is a random experiment with exactly two mutually exclusive outcomes, conventionally labeled "success" and "failure", where the probability of success remains constant.
How do you calculate "at least k" successes?
To find P(X ≥ k), sum the probabilities of obtaining k, k+1, ..., up to n successes, or calculate 1 - P(X ≤ k - 1).
When should I use Poisson instead of Binomial?
Use the Poisson distribution when events occur continuously over time or space with a known average rate λ, or as an approximation for binomial experiments where n is very large (n > 100) and p is very small (p < 0.01).
What is the mean of a binomial distribution?
The mean (expected value) is μ = n × p. For example, rolling a die 60 times with p = 1/6 for a six gives an expected value of 60 × (1/6) = 10 sixes.
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