Bernoulli Distribution

The fundamental building block of discrete probability, modeling a single trial with two outcomes.

The "Single Coin Flip"

The Bernoulli distribution is the simplest of all discrete distributions. It models a single event or trial that has only two possible outcomes: a "success" or a "failure".

Think of it as a single coin flip (Heads or Tails), a single trade (Win or Loss), or a single bond (Default or No Default). The entire distribution is described by a single parameter, pp, which is the probability of success.

The Formula
The probability mass function (PMF) is:
P(X=k)=pk(1−p)1−kfor k∈{0,1}P(X=k) = p^k (1-p)^{1-k} \quad \text{for } k \in \{0, 1\}
  • If k=1k=1 (success), the formula becomes pp.
  • If k=0k=0 (failure), the formula becomes 1−p1-p.
Interactive Bernoulli Trial
Adjust the probability of success (pp) to see how it affects the outcome probabilities.