Bernoulli Distribution Calculator
Compute probabilities and key distribution statistics for a Bernoulli random variable.
A Bernoulli model is one of the simplest and most useful probability tools in statistics. If your experiment has only two outcomes (success/failure, yes/no, pass/fail, click/no click), you can model it with a Bernoulli random variable. This calculator helps you do that quickly and accurately.
What is a Bernoulli random variable?
A Bernoulli random variable X takes only two values:
- X = 1 with probability p (success)
- X = 0 with probability 1 - p (failure)
That’s it. One trial, two outcomes, one parameter.
P(X = x) = px(1 - p)1 - x, for x ∈ {0, 1}
What this Bernoulli calculator gives you
When you enter a valid probability p, this tool computes:
- Probability of success: P(X=1) = p
- Probability of failure: P(X=0) = 1-p
- Optional point probability for chosen x (0 or 1)
- Mean (expected value): E[X] = p
- Variance: Var(X) = p(1-p)
- Standard deviation: √(p(1-p))
- Success odds: p/(1-p)
How to use it
Step 1: Enter probability of success
Use either decimal format (like 0.3) or percent format (like 30%). The tool accepts both.
Step 2: (Optional) Enter an outcome x
If you enter x = 0 or x = 1, the calculator will also return P(X=x) using the Bernoulli PMF formula.
Step 3: Click Calculate
You’ll get a compact summary of all key Bernoulli outputs.
Quick examples
Example 1: Coin flip (biased)
If the probability of heads is p = 0.6:
- P(X=1) = 0.6
- P(X=0) = 0.4
- Mean = 0.6
- Variance = 0.24
Example 2: Email open event
Suppose a campaign has open probability p = 25% (0.25). For one recipient:
- Probability they open = 0.25
- Probability they don’t open = 0.75
- Expected opens per recipient = 0.25
Bernoulli vs Binomial
This is a common point of confusion:
- Bernoulli: one trial
- Binomial: many independent Bernoulli trials
If you repeat the same Bernoulli experiment n times, the total number of successes follows a Binomial distribution.
Common mistakes to avoid
- Entering p outside [0,1] (or outside 0% to 100%)
- Using outcomes other than 0 or 1 for a Bernoulli variable
- Confusing probability with odds
- Applying Bernoulli to problems with more than two outcomes
Why this matters
Bernoulli modeling appears everywhere: product analytics, A/B testing, medical diagnosis, quality control, and finance risk events. Once you understand Bernoulli distributions, more advanced models become much easier to learn.