IB Math AA HL — Normal distribution

Quick example

1. Question — Lamp lifetimes: mean 1200 hours, SD 100 hours. Chance of lasting below 1000 hours?
2. Calculate — normalcdf(−1E99,1000,1200,100) ≈ 0.0228. About 2.28%.

IB-style question — a probability and an expected number (Paper 2)

The masses of oranges are modelled by X ~ N(180, 15²) grams. An orange is rejected if it is lighter than 160 g. (a) Find P(X < 160). (b) In a crate of 500 oranges, find the expected number rejected.

Step by step:

1. The second number is the variance, so σ = √225 = 15.

   $$X \sim N(180,\ 15^{2})$$

2. P(X < 160): lower bound −1ᴇ99, upper 160.

   $$P(X < 160) = \text{normalcdf}(-1\text{E}99,\ 160,\ 180,\ 15) \approx 0.0912$$

3. Expected number = probability × total.

   $$500 \times 0.0912 \approx 46$$

IB-style question — symmetry and comparing curves (Paper 1)

Two classes sit the same test. Both sets of marks are normal with mean 62, but class A has σ = 5 and class B has σ = 12. (a) Write down P(mark > 62) for class A. (b) Whose marks are more consistent, and what does the curve look like?

Step by step:

1. 62 is the mean, and the curve is symmetric about it.

   $$P(\text{mark} > 62) = 0.5$$

2. Smaller σ means less spread — taller and narrower.

   $$\sigma_A = 5 < \sigma_B = 12$$