The idea

Finding normal probabilities

Finding P(X < a), P(X > a) and P(a < X < b) for a normal distribution as areas under the curve, and the inverse problem — finding the value x for a given probability (the inverse normal).

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Statistics · Statistical distributions

Finding normal probabilities

Finding P(X < a), P(X > a) and P(a < X < b) for a normal distribution as areas under the curve, and the inverse problem — finding the value x for a given probability (the inverse normal).

Why it works

Probabilities are areas

Exam marks are distributed

X \sim N(50, 16)

— what proportion of students scored over 56? There's nothing to count: a probability for a normal variable is an area under the bell curve. Modern calculators give these areas directly from

\mu

and

\sigma

, but the thinking is always about which area you want — like the shaded left-region below.

The three shapes of question

$P(X < a)$ is the area to the left of $a$ (the shaded region above, with the curve standardised).
$P(X > a) = 1 - P(X < a)$ — the area to the right is the complement.
$P(a < X < b) = P(X < b) - P(X < a)$ — subtract the smaller left-area from the larger.

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