3.2StatisticsCore

Finding Probabilities

Probabilities for a normal distribution are areas under the curve. Modern calculators compute these directly — you enter μ, σ and the interval, and read off the probability.

24 min The Normal Distribution

Edexcel A Level Maths: 3.2 Finding Probabilities for The Normal DistributionCurated by Kepler, not produced by us. Watch it before the notes below.

Open on YouTube

What you'll be able to do

Find P(X < a), P(X > a), P(a < X < b)
Use the calculator’s normal CD function
Exploit symmetry
Interpret probabilities as areas

Probabilities as areas

For a continuous distribution, $P(X < a)$ is the area under the curve to the left of $a$ . Because the distribution is continuous, $P(X = a) = 0$ , so $P(X < a) = P(X \le a)$ .

P(a < X < b) = P(X < b) - P(X < a)

Area between two values.

Example 1For

X \sim N(50, 4^2)

, describe finding

P(X > 54)

1Use the calculator’s normal CD with lower 54, upper ∞ (or a large number).

2Or

1 - P(X < 54)

Answer

About 0.1587 (1σ above the mean).

Tip — P(X > a) = 1 − P(X < a); use symmetry to check answers.

Formula recap

P(X > a) = 1 - P(X < a)

Complement.

P(a < X < b) = P(X<b) - P(X<a)

Interval.

Common mistakes to avoid

Worrying about < vs ≤ for a continuous variable.

P(X = a) = 0, so they are equal.

Entering σ² instead of σ into the calculator.

Calculators take the standard deviation σ.

Key takeaways

Probabilities are areas under the curve.
P(X > a) = 1 − P(X < a); intervals subtract.
Calculators take μ and σ (not σ²).

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

For a continuous distribution, P(X = a) is:
- $0$
- $1$
- $0.5$
- undefined
Show answer and explanation
Answer: $0$
Single point.
$P(X > a)$ equals:
- $1 - P(X < a)$
- $P(X < a)$
- $P(X = a)$
- $1$
Show answer and explanation
Answer: $1 - P(X < a)$
Complement.
A probability for a normal distribution represents:
- an area under the curve
- a height
- a gradient
- a count
Show answer and explanation
Answer: an area under the curve
Continuous distribution.
$P(a < X < b)$ equals:
- $P(X<b) - P(X<a)$
- $P(X<a) - P(X<b)$
- $P(X<a)P(X<b)$
- $P(X<b)$
Show answer and explanation
Answer: $P(X<b) - P(X<a)$
Subtract areas.
A calculator’s normal function takes:
- μ and σ
- μ and σ²
- only σ
- n and p
Show answer and explanation
Answer: μ and σ
Standard deviation.

Next step

Check your understanding

Work through questions on Finding Probabilities. Use the feedback to decide what needs another look.

6 past-paper questions · 10 marks

Practise this topic

Related lessons

All 7 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.