Bernoulli Distribution

These two outcomes are mathematically coded as:

• Success ($1$)
• Failure ($0$)

Core Concepts

If a random variable $X$ follows a Bernoulli distribution, we write this as:

Where $p$ is the probability of the "Success" outcome. Therefore, the probability of the "Failure" outcome is naturally $1 - p$ (sometimes denoted as $q$).

Probability Mass Function (PMF)

The probability mass function is piecewise defined as:

This can be written compactly into a single elegant algebraic formula for $x \in \{0, 1\}$:

Key Metrics

• Expected Value (Mean): $E[X] = p$
• Variance: $\text{Var}(X) = p(1 - p)$

Real-World Examples

1. Flipping a Coin: If we define "Heads" as success, a fair coin has $p = 0.5$.
2. Quality Control: Testing a single lightbulb where success is "defective" ($p = 0.02$) and failure is "working".
3. Medical Testing: A patient either tests positive for a disease ($1$) or negative ($0$).

Interactive Visualization

Below is an interactive graph. Notice how increasing the probability of success $p$ physically shifts the weight from the "Failure" bar directly over to the "Success" bar. Because there are only two outcomes, the sum of the two bars will always equal exactly $1.0$.

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