Bernoulli trials | Year 12

01 / LEARN

The idea, step by step

Success is a label, not necessarily a good result

A Bernoulli trial has two categories: success and failure. Choose which event is called success before assigning its probability p. If failure means a faulty item, it may still be convenient to call a fault a success for the purpose of counting faults.

Define an indicator $B = 1$ when the selected event occurs and $B = 0$ otherwise. Then $P (B = 1) = p$ and $P (B = 0) = 1 - p$ . One Bernoulli variable describes one trial, not the total number of successes in a larger experiment.

Derive, rather than memorise, the summaries

$E (B) = 0 (1 - p) + 1 p = p$ . Because both possible values satisfy $B^{2} = B$ , $E (B^{2}) = p$ too. Hence $Var (B) = p - p^{2} = p (1 - p)$ .

The variance is largest when success and failure are equally likely, and is zero when the outcome is certain. A payoff of 5 dollars for success is 5B, not a Bernoulli variable itself; its mean and variance follow the scale rules.

A common trap

Every experiment with two numerical outcomes is itself Bernoulli.

Instead: A Bernoulli variable uses the values 0 and 1. A two-outcome payoff can be represented by a transformed Bernoulli indicator.

02 / WORKED EXAMPLES

See the thinking

Example 1

An event called success

A sensor passes a check with probability 0.65. Define B = 1 for a pass. Find its mean and variance.

The distribution is P(B = 1) = 0.65 and P(B = 0) = 0.35.
E(B) = p = 0.65.
Var(B) = p(1 − p) = 0.65 × 0.35 = 0.2275.

Mean 0.65; variance 0.2275.

Example 2

Turn a count event into an indicator

X is 0, 1 or 2 with probabilities 0.4, 0.35 and 0.25. Let B = 1 if X ≥ 1 and 0 otherwise. Find the distribution and variance of B.

Success is the event X ≥ 1, so p = 0.35 + 0.25 = 0.6.
Therefore B is 1 with probability 0.6 and 0 with probability 0.4.
Var(B) = 0.6 × 0.4 = 0.24.

B has probabilities 0.4, 0.6 at 0, 1; variance 0.24.

Check your understanding

For B Bernoulli with p = 0.2, find P(B = 0).

Show answer

0.8.

The two probabilities must sum to 1.

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A coin lands heads with probability 0.55. Define a Bernoulli indicator for tails and give p.
2 marks
B is Bernoulli with p = 0.4. Find E(B), Var(B) and SD(B).
3 marks
Let B be Bernoulli with p = 0.1. A bonus is Y = 20B dollars. Find its mean and variance.
3 marks
If B is Bernoulli with p = 1, describe B and find its variance.
2 marks
A variable equals 2 or 7. Explain how a Bernoulli variable B can represent it.
2 marks
Show that a Bernoulli variance cannot exceed 1/4.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A coin lands heads with probability 0.55. Define a Bernoulli indicator for tails and give p.
2 marks
The chosen success event is tails.
Its probability is 1 − 0.55.
B = 1 for tails, 0 for heads; p = 0.45.
B is Bernoulli with p = 0.4. Find E(B), Var(B) and SD(B).
3 marks
Use E(B) = p.
Use Var(B) = p(1 − p) and take its square root.
$0.4; 0.24; \sqrt{0.24} \approx 0.4899$ .
Let B be Bernoulli with p = 0.1. A bonus is Y = 20B dollars. Find its mean and variance.
3 marks
E(Y) = 20(0.1) = 2.
Var(Y) = 20²(0.1)(0.9) = 36.
Mean 2 dollars; variance 36 dollars².
If B is Bernoulli with p = 1, describe B and find its variance.
2 marks
Success is certain.
The formula p(1 − p) gives 1 × 0 = 0.
B is always 1; variance 0.
A variable equals 2 or 7. Explain how a Bernoulli variable B can represent it.
2 marks
If B = 0 the expression is 2.
If B = 1 it is 7; p is the probability of the outcome 7.
X = 2 + 5B, with B = 1 for the outcome 7.
Show that a Bernoulli variance cannot exceed 1/4.
3 marks
Complete the square in p − p².
A square is non-negative, with equality in the bound when p = 1/2.
$p (1 - p) = 1 / 4 - {(p - 1 / 2)}^{2} \leq 1 / 4$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods: discrete random variables and probability distributions, expected value and variance, transformations and independent sums, Bernoulli and binomial models. Original supporting lessons and exercises follow the approved Ascent chapter structure.

Official curriculum source ↗

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