← Probability & Distributions
Mathematics · Topic Cheatsheet

Probability & Distributions

19 key results accumulated across 2 chapters.

Basic probability
Ch 1
P(A)=favourabletotalP(A) = \frac{\text{favourable}}{\text{total}}
Complement
Ch 1
P(A′)=1−P(A)P(A') = 1 - P(A)
Union
Ch 1
P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
Conditional
Ch 1
P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}
Independence
Ch 1
P(A∩B)=P(A) P(B)P(A \cap B) = P(A)\,P(B)
Combining events
Ch 1
P(A∪B)=P(A)+P(B)−P(A∩B)P(A\cup B)=P(A)+P(B)-P(A\cap B)
Conditional / independent
Ch 1
P(A∣B)=P(A∩B)P(B);  indep:P(A∩B)=P(A)P(B)P(A|B)=\tfrac{P(A\cap B)}{P(B)};\;\text{indep}: P(A\cap B)=P(A)P(B)
Complement & tree
Ch 1
P(at least one)=1−P(none)P(\text{at least one})=1-P(\text{none})
Multiply along branches; add across outcomes.
Common trap
Ch 1
Mutually exclusive (can’t both happen) ≠ independent (one doesn’t affect the other).
Binomial
Ch 2
X∼B(n,p):  P(X=x)=(nx)px(1−p)n−xX\sim B(n,p):\; P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}
Binomial mean / variance
Ch 2
E(X)=np,Var(X)=np(1−p)E(X)=np,\quad \mathrm{Var}(X)=np(1-p)
Normal
Ch 2
X∼N(μ,σ2)X\sim N(\mu,\sigma^2)
Symmetric bell about μ\mu; total area = 1.
Standardisation
Ch 2
Z=X−μσ∼N(0,1)Z=\frac{X-\mu}{\sigma}\sim N(0,1)
Empirical rule
Ch 2
≈68% within μ±σ\mu\pm\sigma, ≈95% within ±2σ\pm2\sigma, ≈99.7% within ±3σ\pm3\sigma.
Inverse normal
Ch 2
x=μ+zσx=\mu+z\sigma
Find zz from the required probability, then scale back.
Binomial
Ch 2
X∼B(n,p),  E(X)=np,  Var=np(1−p)X\sim B(n,p),\; E(X)=np,\; \text{Var}=np(1-p)
Fixed n trials, constant p, independent.
Normal
Ch 2
Symmetric bell; use GDC for P(a
Expected value
Ch 2
E(X)=∑x P(x)E(X)=\sum x\,P(x)
Common trap
Ch 2
Binomial is discrete (use =), normal is continuous (P(X=x)=0); inverse-normal for ‘find x given probability’.