1. The Bernoulli Scheme: Definition and Three Hypotheses
Core Idea: A Bernoulli scheme (or sequence of Bernoulli trials) is a finite sequence of experiments that satisfies three strict conditions. When all three hold, we unlock the powerful binomial probability formula.
The Three Hypotheses
H1: Two Outcomes
Each trial produces exactly two outcomes, labelled success (S) and failure (F). These are mutually exclusive and exhaustive.
Example: A coin flip — Heads or Tails. A manufactured part — Defective or Good.
H2: Constant Probability
The probability of success $p$ is the same on every trial. We write the failure probability as:
$$q = 1 - p$$
This notation is fixed for the entire topic.
H3: Independence
The outcome of any trial does not affect the outcome of any other trial. Knowing what happened on trial 1 tells you nothing about trial 2.
What Breaks Each Hypothesis?
| Hypothesis | Violation Scenario | Why It Breaks |
|---|---|---|
| H2 — Constant $p$ | A student practising free throws improves over time (learning effect) | The success probability changes from trial to trial, so $p$ is not constant. |
| H3 — Independence | Drawing cards from a deck without replacement | Removing a card changes the composition of the deck, so the probability on the next draw depends on what was drawn before. |
| H2 — Constant $p$ | Manufacturing yield varies across shifts due to weather or fatigue | Environmental changes make $p$ drift, violating constancy. |
| H3 — Independence | Contagious disease testing in a household | One person's infection raises the probability for others — outcomes are correlated. |
Entry Skill — Recognising Bernoulli Setups: Before reaching for the formula, always check all three hypotheses against the story. If any one fails, the binomial formula does not apply. This recognition step is the gatekeeper for every problem in this topic.