The Bernoulli trial and scheme

Many situations reduce to the same structure: an experiment with only two outcomes is repeated, and the successes are counted. It is the most frequent scheme in all of applied probability.

The Bernoulli trial

A Bernoulli trial is a random experiment with only two outcomes, conventionally called "success" and "failure".

The parameter p denotes the probability of success; that of failure is therefore 1 - p, often written q.

Experiment                        Success             p
--------------------------------  ------------------  --------
tossing a coin                    getting heads       0.5
rolling a die                     getting a 6         1/6
inspecting a machined part        part is defective   0.02
polling a voter                   answer "yes"        unknown

The word "success" carries no value judgement: in quality control, the "success" is often the defect. It is simply the outcome being counted.

If X is 1 on a success and 0 otherwise:

E(X) = 1×p + 0×(1-p) = p
V(X) = p(1-p)

The variance is maximal at p = 0.5 — uncertainty is greatest when both outcomes are equally likely — and zero at p = 0 or p = 1, where the result is certain.

The Bernoulli scheme

Now repeat the same trial n times, under the same conditions and independently. This is a Bernoulli scheme with parameters n and p.

The three conditions must be checked systematically:

1. each trial has only TWO outcomes
2. the probability p is THE SAME at each repetition
3. the trials are INDEPENDENT

The tree

For n = 3 repetitions the tree has 2³ = 8 paths:

                  /-- S   SSS
           /-- S <
          /       \-- F   SSF
   /-- S <
  /       \        /-- S   SFS
 /         \-- F <
S                  \-- F   SFF
 \        /-- S ...  FSS, FSF
  \-- F <
          \-- F ...  FFS, FFF

Each path has a probability obtained by multiplying those of its branches — independence is what allows this:

P(SSF) = p × p × (1-p) = p²(1-p)
P(SFS) = p × (1-p) × p = p²(1-p)        the same value!
P(FSS) = (1-p) × p × p = p²(1-p)

The decisive observation: all paths with the same number of successes have the same probability, whatever the order. So all that remains is to count the paths — precisely the role of combinations.

When the scheme does not apply

Draw WITHOUT replacement from an urn  ->  p changes each draw (condition 2)
Trials influencing one another        ->  independence broken (condition 3)
More than two outcomes                ->  condition 1

A useful nuance: when drawing without replacement from a very large population (a survey of 1000 people among 40 million), the composition varies so little that the draw is treated as one with replacement. This is what licenses handling surveys with the binomial distribution.

Summary

• A Bernoulli trial has only two outcomes, with probabilities p and 1 - p.
• "Success" is simply the outcome being counted, with no value judgement.
• A Bernoulli scheme repeats that trial n times, with constant p and independently.
• In the tree, all paths with k successes have the same probability p^k(1-p)^(n-k).
• The three conditions — two outcomes, constant p, independence — must be checked before any computation.
• A draw without replacement from a very large population behaves like one with replacement.