Normal probabilities

Objective. Calculate intervals and tails accurately.

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Learn: the key idea

Normal probabilities use standardisation and the standard normal cumulative distribution function Φ(z) to find the probability a variable falls below, above, or between given values. P(X < x) = Φ(z), P(X > x) = 1 − Φ(z), and P(a < X < b) = Φ(z_b) − Φ(z_a), where each z is the standardised score. For X ~ N(20, 25), P(X < 25) uses z = (25−20)/5 = 1, so P(X < 25) = Φ(1) ≈ 0.8413. Always subtract from 1 for 'greater than' and subtract two Φ values for an interval.

Worked example

Given X ~ N(30, 16), find the normal probability P(X > 34), using Φ(1) = 0.8413.

1 $Standardise: z = (34-30)/4 = 1$
2 $Find Φ(1) = 0.8413, representing P(X < 34)$
3 $Apply P(X > 34) = 1 - Φ(1)$
4 $Compute 1 - 0.8413 = 0.1587$

Answer: 0.1587

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Common mistake. A frequent error is forgetting to subtract from 1 when finding P(X > x), instead reporting Φ(z) directly as the upper-tail probability. Always compute 1 − Φ(z) for 'greater than' probabilities.

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