Binomial Distribution Calculator
Calculate binomial probabilities, visualize distributions, and understand the likelihood of successes.
Total number of experiments.
Likelihood of success in one trial (0 to 1).
Desired number of successful outcomes.
Calculation Result:
The probability of getting successes in trials is:
Binomial Distribution Visualization
Understanding Binomial Distribution
The Binomial Distribution is a cornerstone of probability, describing the likelihood of achieving a specific number of successes in a series of independent experiments, known as trials. Each trial has only two outcomes: success or failure.
Key Concepts:
- Trials (n): The total count of repeated experiments. Each trial is independent.
- Success Probability (p): The constant probability of success in each trial, remaining consistent across all trials.
- Successes (k): The number of successful outcomes for which we want to calculate the probability.
Formula: The probability of exactly k successes in n trials is given by: $$P(X=k) = {n \choose k} \cdot p^k \cdot (1-p)^{n-k}$$ where ${n \choose k}$ is the binomial coefficient, representing the number of ways to choose k successes from n trials.
Applications: Binomial distribution is used across many fields:
- Quality Control: To estimate defective product rates.
- Medical Research: To assess treatment success probabilities.
- Surveys and Polling: To analyze the likelihood of specific response numbers.
- Genetics: For modeling trait inheritance patterns.
For deeper understanding, explore the Wikipedia article on Binomial Distribution.