Bernoulli Distribution Calculator
Enter a probability of success p and choose the outcome x (0 or 1). The calculator returns the probability mass value and key distribution metrics.
What Is a Bernoulli Distribution?
A Bernoulli distribution is the simplest discrete probability distribution. It models an experiment with only two outcomes: success (1) and failure (0). If the probability of success is p, then the probability of failure is 1 - p.
Typical examples include flipping a biased coin (heads = 1, tails = 0), whether a user clicks an ad (click = 1), or whether a manufactured part passes inspection (pass = 1).
Probability Mass Function (PMF)
P(X = 1) = p
P(X = 0) = 1 - p
Since there are only two outcomes, calculating probabilities is direct and intuitive.
Key Properties
- Support: X ∈ {0, 1}
- Mean (Expected Value): E[X] = p
- Variance: Var(X) = p(1 - p)
- Standard Deviation: √(p(1 - p))
How to Use This Calculator
Step 1: Enter p
Input a valid probability between 0 and 1. For example, p = 0.2 means success happens 20% of the time.
Step 2: Enter x
Enter x as either 0 or 1. Any other value is invalid for a Bernoulli random variable.
Step 3: Calculate
Click the calculate button to get:
- P(X = x) for your chosen outcome
- Mean, variance, and standard deviation
- CDF value at x
Worked Example
Suppose a player makes a free throw with probability p = 0.75.
- Probability of making the shot (x = 1): P(X = 1) = 0.75
- Probability of missing (x = 0): P(X = 0) = 0.25
- Expected value: E[X] = 0.75
- Variance: 0.75 × 0.25 = 0.1875
Bernoulli vs. Binomial
A Bernoulli random variable represents one trial. A binomial random variable represents the sum of many independent Bernoulli trials with the same p.
- Bernoulli: one yes/no trial
- Binomial: number of successes across n trials
Common Mistakes to Avoid
- Using p outside the interval [0, 1]
- Entering x values other than 0 or 1
- Confusing Bernoulli (single trial) with binomial (multiple trials)
- Forgetting that P(X=0) is 1-p, not p
Why This Distribution Matters
The Bernoulli model appears everywhere: machine learning labels, conversion events in marketing, reliability testing, quality control, and medical outcomes (success/failure). Understanding it builds a foundation for more advanced models such as binomial, geometric, and logistic regression frameworks.