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 Video by Zeeshan Zamurred The Normal Distribution

Edexcel A Level Maths: 3.2 Finding Probabilities for The Normal DistributionWatch the full walkthrough before the notes below.

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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 $P (X < a) = P (X \leq a)$ .

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

Area between two values.

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

2Or

1 - P (X < 54)

AnswerAbout 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).

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