Mini-Lesson

Statistics & Probability (HL)

This HL mini-lesson strengthens the statistics core: conditional probability and Bayes' theorem, the binomial and normal distributions, and the χ² test for independence — all interpreted carefully in context. (Poisson, further hypothesis tests and Markov chains are in the dedicated HL Statistics lesson.)

AI HL flavour: HL adds Bayesian reasoning and demands precise interpretation of tests and distributions.

Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect ⭐ stars. Every number on the calculation screens has been re-derived and checked. Press Start when you are ready.

Probability

Conditional probability

Conditional probability updates a probability given new information:

P(A | B) = P(A ∩ B) ÷ P(B)independent ⇔ P(A | B) = P(A)

Read two-way tables carefully: the condition B restricts you to one row or column.

Bayes

Bayes' theorem

Bayes' theorem reverses a conditional probability — vital for diagnostic testing:

P(D | +) = P(+ | D)P(D) ÷ P(+)P(+) = P(+|D)P(D) + P(+|D')P(D')

Worked example — rare disease

Prevalence P(D) = 0.01, sensitivity P(+|D) = 0.99, specificity 0.95 (so P(+|D') = 0.05).

P(+) = 0.99×0.01 + 0.05×0.99 = 0.0594

P(D | +) = 0.99×0.01 ÷ 0.0594 = 0.1667 — only about 17%!

Calculate

Your turn — Bayes

1A disease has prevalence 0.01. A test has sensitivity 0.99 and specificity 0.95 (false-positive rate 0.05). Find P(disease | positive) to 4 decimal places.

Hint: numerator 0.99×0.01; denominator 0.99×0.01 + 0.05×0.99.

Quick check

Interpreting Bayes

?The rare-disease result P(D | +) ≈ 0.17 shows that:

Binomial

The binomial distribution

X ~ B(n, p) counts successes in n independent trials.

P(X = r) = ⁿC_r p^r(1−p)^n−rmean np, variance np(1−p)

Worked example

X ~ B(10, 0.3): P(X = 4) = 0.2001, mean = 3, P(X ≤ 2) = 0.3828.

Calculate

Your turn — binomial

2For X ~ B(10, 0.3), find P(X = 4) to 4 decimal places.

Hint: ¹⁰C₄ × 0.3⁴ × 0.7⁶.

Normal

The normal distribution

X ~ N(μ, σ²) is the bell curve; probabilities are areas (GDC normalcdf / inverse normal).

Worked example

X ~ N(50, 8²): P(X < 60) = 0.8944; 90th percentile = 60.25.

Quality control X ~ N(500, 20²): P(X > 530) = 0.0668

Calculate

Your turn — normal

3For X ~ N(500, 20²), find P(X > 530) to 4 decimal places.

Hint: z = (530 − 500) ÷ 20 = 1.5; find the upper-tail area.

χ² test

Chi-square test for independence

Test whether two categorical variables are independent.

χ²_calc = Σ (O − E)² ÷ Edf = (r − 1)(c − 1)

Worked example

2×3 table gives χ²calc = 16.67, df = 2, critical value 5.991.

16.67 > 5.991 ⇒ reject H₀: the variables are associated.

Calculate

Your turn — degrees of freedom

4A χ² test uses a 3 × 3 table. State the degrees of freedom (r − 1)(c − 1).

Hint: (3 − 1)(3 − 1).

Calculate

Your turn — χ² decision

5A χ² test gives χ²calc = 16.67 with critical value 5.991. Enter 1 to reject H₀, or 0 to not reject.

Hint: reject H₀ when χ²calc exceeds the critical value.

Sort it

Probability or distribution?

Tap a term, then the strand it belongs to.

🎯 Conditional / Bayes

🔔 Distribution

🧪 Hypothesis test

Quick check

χ² expected values

?In a χ² test, each expected value E is computed as:

Quick check

Independence check

?Events A and B are independent if and only if:

Match it

Match the statistics tool to its formula

Tap an item on the left, then its match on the right.

Item

Match

Expectation

Expected value

The expected value E(X) = Σ x·P(X = x) is the long-run average outcome — central to fair games and risk.

Worked example

A game pays $5 with probability 0.2, else $0.

E(X) = 5 × 0.2 + 0 × 0.8 = $1.00 per play

Quick check

Fair game

?A game costs $1 to play and has expected winnings E(X) = $1.00. The game is:

Recap

The big ideas to take away

Conditional: P(A|B) = P(A∩B) ⁄ P(B)

Bayes: P(D|+) reverses the conditional; beware rare-event false positives

Binomial: B(n,p): mean np, variance np(1−p)

Normal: N(μ,σ²): GDC areas and inverse normal

χ² test: Σ(O−E)²⁄E, df=(r−1)(c−1); E = row×col ⁄ total

Interpretation: state H₀/H₁, compare to critical value or p-value

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