Bernoulli Trial

Formula

\[ X \sim \mathrm{Bernoulli}(p),\quad X\in\{0,1\} \]

\[ \hat p = \frac{1}{n}\sum_{i=1}^n X_i \]

Parameters

• \(p\): success probability
• \(X_i\): trial outcomes
• \(\hat p\): sample estimate of \(p\)

What it means

A single random experiment with two outcomes (success/failure).

What it's used for

• Modeling binary events.
• Estimating a success probability from samples.

Key properties

• \(\hat p\) is the sample mean of outcomes.
• Repeated trials lead to a binomial count.

Common gotchas

• Outcomes must be independent for standard estimators.
• Be explicit about what counts as "success".

Example

If 3 successes in 10 trials, \(\hat p=0.3\).

How to Compute (Pseudocode)

Input: success probability p and number of trials n (or observed binary outcomes)
Output: Bernoulli trial simulation/outcome summary

for i from 1 to n:
  sample X_i in {0,1} with P(X_i=1)=p   # or record observed outcomes
compute summary statistics such as sample proportion if needed
return outcomes (and summaries)

Complexity

• Time: \(O(n)\) for \(n\) simulated/observed trials
• Space: \(O(n)\) to store all outcomes (or \(O(1)\) extra if only accumulating counts)
• Assumptions: Independent Bernoulli-trial workflow shown; exact PMF/probability formulas are separate computations