IB Maths AA SL Topic 4 — Stats & Probability Paper 2 🎯 Skill ~3 min practice

AA SL Normal Probability skills

For continuous data that bunches around a mean — heights, test scores, weights, errors. Two GDC functions handle everything: normalCdf for finding a probability, invNorm for finding the value. Pick the right one and these are 30-second wins.

The Method

X ~ N(μ, σ²) μ = mean · σ = standard deviation · σ² = variance

Function 1

P(a < X < b)

normalCdf(a, b, μ, σ)

probability between two values

Function 2

P(X < a) or P(X > a)

use −10⁹⁹ or 10⁹⁹ for ∞

tail — replace missing bound with ±∞

Function 3

Find x from probability

invNorm(area, μ, σ)

“top 10%” / “score for 95th percentile”

What each function calculates

The probability is always the shaded area under the curve. Total area under any normal curve = 1, so probabilities are between 0 and 1.

How to find normalCdf / invNorm

TI-84 Plus

Press 2nd → VARS (DISTR menu)
Choose 2: normalcdf( for probability or 3: invNorm( for value
Enter: lower, upper, μ, σ (or area, μ, σ for invNorm)
For ±∞ use −1×10^99 or 1×10^99

Casio fx-CG50

Open STAT menu → DIST (F5)
Choose NORM (F1)
Pick Ncd (F2) for probability, InvN (F3) for value
Set Data: Variable, then enter Lower, Upper, σ, μ

Worked examples

WE 1 EASY

The heights of a group of adults are normally distributed with mean 170 cm and SD 8 cm. Find the probability that a randomly chosen adult is shorter than 180 cm. Give answer to 3 sf.

step 1 — set up X ~ N(170, 8²) — find P(X < 180)step 2 — use lower bound −∞ P(X < 180) = normalCdf(−10⁹⁹, 180, 170, 8)step 3 — read GDC ≈ 0.89435…P(X < 180) ≈ 0.894 (3 sf) “shorter than” = strict less than → use −10⁹⁹ as the lower bound.

WE 2 MEDIUM

Test scores are normally distributed with mean 65 and SD 12. Find the probability of scoring between 50 and 75. Give answer to 3 sf.

step 1 — set up X ~ N(65, 12²) — find P(50 < X < 75)step 2 — both bounds finite P(50 < X < 75) = normalCdf(50, 75, 65, 12)step 3 — GDC ≈ 0.69077…probability ≈ 0.691 (3 sf) “between” with both numerical bounds — easiest case, plug both in.

WE 3 HARD

Lifespans of a battery brand are normally distributed with mean 600 hours and SD 50 hours. The top 5% are labelled “premium”. Find the minimum lifespan needed to be premium, to 3 sf.

step 1 — interpret “top 5%” need x where P(X > x) = 0.05 → P(X < x) = 0.95 (the 95th percentile)step 2 — use invNorm with area = 0.95 x = invNorm(0.95, 600, 50)step 3 — GDC ≈ 682.24…minimum lifespan ≈ 682 hours (3 sf) invNorm always uses area to the LEFT — for “top X%”, use 1 − X% as the area.

Practice questions

Try each one yourself first, then click the question to reveal the worked answer. For invNorm questions, sketch the bell curve to spot which area you need.

Q1 EASY X ~ N(50, 10²). Find P(X < 65), to 3 sf. Show answer ▼Hide answer ▲

normalCdf(−10⁹⁹, 65, 50, 10) ≈ 0.93319… P(X < 65) ≈ 0.933 (3 sf)

Q2 EASY X ~ N(100, 15²). Find P(X > 110), to 3 sf. Show answer ▼Hide answer ▲

normalCdf(110, 10⁹⁹, 100, 15) ≈ 0.25249… P(X > 110) ≈ 0.252 (3 sf)

Q3 MEDIUM Weights of apples ~ N(200 g, 15² g²). Find P(180 < X < 220), to 3 sf. Show answer ▼Hide answer ▲

normalCdf(180, 220, 200, 15) ≈ 0.81758… P(180 < X < 220) ≈ 0.818 (3 sf)

Q4 MEDIUM Test scores ~ N(70, 8²). The top 25% receive a distinction. Find the minimum distinction score, to 3 sf. Show answer ▼Hide answer ▲

“top 25%” → 75th percentile x = invNorm(0.75, 70, 8) ≈ 75.395… minimum ≈ 75.4 (3 sf) always convert “top X%” to “area to the left” — invNorm wants left area.

Q5 HARD X ~ N(50, 6²). Find k such that P(k < X < 56) = 0.4, to 3 sf. Show answer ▼Hide answer ▲

step 1 — find P(X < 56) normalCdf(−10⁹⁹, 56, 50, 6) ≈ 0.8413step 2 — area to left of k P(X < k) = 0.8413 − 0.4 = 0.4413step 3 — invNorm k = invNorm(0.4413, 50, 6) ≈ 49.115… k ≈ 49.1 (3 sf) when P(k < X < b) is given, subtract from P(X < b) to get P(X < k), then invNorm.

⚠ Common mistakes

Entering σ² instead of σ. The GDC needs the standard deviation, not the variance. If question says variance = 64, enter σ = 8.
Wrong area for invNorm. invNorm uses area to the left. For “top 10%”, use area = 0.90, not 0.10.
Forgetting ±∞ for tails. P(X < a) needs lower = −10⁹⁹. P(X > a) needs upper = 10⁹⁹. Most calculators won’t accept just leaving it blank.
Using normalCdf when invNorm is needed. If the question asks for a value given a probability, use invNorm.
Mixing up μ and σ in the input. TI-84 takes (lower, upper, μ, σ). Casio takes (Lower, Upper, σ, μ). Order matters.
Strict vs not-strict inequality. For continuous distributions, P(X < a) = P(X ≤ a) — they’re the same. Don’t lose marks worrying about it.

📖

Want the theory?

Read the full Normal Distribution notes for the bell curve properties, the link to z-scores, and how the empirical 68–95–99.7 rule lets you sanity-check answers.

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