This HL mini-lesson extends probability: conditional probability, independence, the law of total probability, Bayes' theorem, expectation and variance, and continuous random variables via probability density functions.
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Conditional
Conditional probability
The probability of A given B is:
P(A | B) = P(A ∩ B) / P(B)rearranges to P(A ∩ B) = P(A|B)·P(B)
Quick check
Bayes
?Bayes’ theorem is used to:
Independence
Independence & mutual exclusivity
Independent: P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A). Mutually exclusive: P(A ∩ B) = 0.
Don't confuse them: mutually exclusive events (except trivial ones) are not independent — if one happens the other cannot.
Calculate
Bayes
✎A disease has P(D)=0.01. A test gives P(+|D)=0.99 and P(+|D′)=0.05. Find P(D|+) to 3 d.p.
Hint: P(+) = 0.99(0.01) + 0.05(0.99); divide 0.0099 by it.
Total probability
The law of total probability
If B₁, B₂, … partition the sample space, then for any event A:
P(A) = Σ P(A | Bᵢ)·P(Bᵢ)the denominator in Bayes' theorem
Quick check
Continuous variable
?For a continuous random variable X, P(X = a) for a single value a equals:
Calculate
Conditional
✎Given P(A ∩ B) = 0.12 and P(B) = 0.3, find P(A | B).
✎f(x) = kx is a pdf on 0 ≤ x ≤ 2. Find k so the total area is 1.
Hint: ∫₀² kx dx = k·2 = 1.
Quick check
Total probability
?If B₁,…,Bₙ partition the sample space, P(A) equals:
Continuous
Continuous random variables
A continuous variable has a probability density function f(x) with total area 1. Probabilities and expectation are integrals:
∫ f(x) dx = 1 · E(X) = ∫ x·f(x) dxand P(X = a) = 0 for any single value a
Calculate
Continuous mean
✎For f(x) = 0.5x on [0, 2], find E(X) = ∫₀² x·f(x) dx to 3 d.p.
Hint: 0.5 ∫₀² x² dx = 0.5 × 8/3.
Match it
Match each quantity to its formula
Tap a statement on the left, then its matching partner on the right.
Quantity
Formula
Distributions
Binomial & normal at HL
Binomial X ~ B(n, p): P(X = r) = ₙCᵣ pʳ(1−p)ⁿ⁻ʳ, E(X) = np, Var(X) = np(1−p). The normal N(μ, σ²) is standardised with z = (x − μ)/σ.
Calculate
Variance
✎A variable has E(X) = 2 and E(X²) = 6. Find Var(X).
Hint: Var(X) = E(X²) − [E(X)]² = 6 − 4.
Quick check
Variance
?Variance can be computed as:
Strategy
Reasoning with probability
Draw a tree or table for two-stage experiments; use total probability for the denominator and Bayes to invert a condition. For continuous variables, integrate the density.
Exam habit: check probabilities sum to 1 and lie in [0, 1]; a Bayes answer should respect the base rate.
Recap
The big ideas to know
Conditional: P(A|B)=P(A∩B)/P(B); independent if =P(A)
Total probability: P(A)=Σ P(A|Bᵢ)P(Bᵢ)
Bayes: P(D|+)=P(+|D)P(D)/P(+)
Expectation: E(X)=Σ xP(x); Var=E(X²)−E(X)²
Continuous: ∫f=1, E(X)=∫xf dx, P(X=a)=0
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