A shaded region on a normal curve is a picture of the probability you want. Build your expression from the picture, then check that the size of your answer matches how much of the curve is shaded.
This lesson is part of choosing the right probability model. The tail skills are taught in reading normal-distribution tail probabilities.
How does each shaded shape become an expression?
The whole curve has area 1. Use the right tail Q(z) from the table and build everything else from it.
| Shaded region | Expression in Q | Rough size |
|---|---|---|
| Right tail beyond a positive z | Q(z) | Small if z is large |
| Left tail below a negative z | Q(a), with a = −z | Small if a is large |
| Everything left of a positive z | 1 − Q(z) | Larger than 0.5 |
| Middle strip across the mean | 1 − Q(z₁) − Q(z₂) | Large if the strip is wide |
The last row says: take the whole curve and remove the two unshaded tails.
Worked example: a middle strip
Here is an original question. The time X minutes a fictional bus takes is normally distributed with mean 40 and standard deviation 5. The diagram shades the region between X = 35 and X = 47. Find the probability.
- Standardise both ends. For 35: z = (35 − 40) ÷ 5 = −1. For 47: z = (47 − 40) ÷ 5 = 1.4.
- Describe the picture. The strip crosses the mean, so it is a large central region. The unshaded parts are a left tail below z = −1 and a right tail above z = 1.4.
- Build the expression: P(35 < X < 47) = 1 − Q(1) − Q(1.4).
- Read the table: Q(1) = 0.1587 and Q(1.4) = 0.0808.
- Calculate: 1 − 0.1587 − 0.0808 = 0.7605.
Size check: the strip runs from 1 standard deviation below the mean to 1.4 above it, so most of the curve is shaded, and 0.76 agrees.
The mistake that costs marks
The common slip is to treat both ends like right tails and subtract them, writing Q(1.4) − Q(1). The working looks like a difference of two table values, so it feels right.
| Step | Wrong | Right |
|---|---|---|
| Expression | Q(1.4) − Q(1) | 1 − Q(1) − Q(1.4) |
| Calculation | 0.0808 − 0.1587 | 1 − 0.1587 − 0.0808 |
| Result | −0.0779 | 0.7605 |
| Size check | Negative, impossible | Large, matches the strip |
A negative probability is a clear signal. A second signal is an answer such as 0.08 for a region that covers the middle of the curve.
The same idea on a one-sided shape
Suppose the diagram instead shades everything to the right of X = 47. The region is a thin right tail, so the answer should be small.
P(X > 47) = Q(1.4) = 0.0808.
If your working gave 0.9192 for this shaded region, you have found the left part instead. The size check shows the mismatch: a thin tail cannot hold 92% of the probability.
Check yourself
X ~ N(30, 4²). The diagram shades the region between X = 28 and X = 36. Find the probability, and say what size you expect before calculating.
Answer
Here σ = 4. For 28: z = (28 − 30) ÷ 4 = −0.5. For 36: z = (36 − 30) ÷ 4 = 1.5.
The strip crosses the mean and covers from 0.5 below to 1.5 above, so expect a large answer, more than 0.5.
P = 1 − Q(0.5) − Q(1.5) = 1 − 0.3085 − 0.0668 = 0.6247.
The answer is large, as expected.
What to study next
The four skills in this cluster come together in mixed questions. Try the integrated practice set, and keep a record of each wrong step with the mistake log and paper-error review.
If you want a teacher to compare your sketch and your working, see online one-to-one Additional Mathematics tuition.