Probability Essentials - Quick Review

Probability basics

Probability ranges between 0 and 1: 0\le P(A)\le 1
Complement rule: P(A^c)=1-P(A)
For equally likely outcomes: P(A)=\frac{\text{# favorable outcomes}}{\text{# total outcomes}}
Classical vs Bayesian interpretations: long-run frequency vs subjective belief

Independent: the outcome of A does not affect B (e.g., coin tosses)
Dependent: the occurrence of A affects the probability of B
General intersection: P(A\cap B)=P(A)\,P(B|A)
If A and B are independent: P(A\cap B)=P(A)\,P(B)
Example: probability of two aces in draws from a 52-card deck: \frac{4}{52}\cdot\frac{3}{51}=\frac{12}{2652}=0.0045

Mutually exclusive: A and B cannot occur together
Addition rule for union: P(A\cup B)=P(A)+P(B)\quad\text{if mutually exclusive}
If not mutually exclusive: P(A\cup B)=P(A)+P(B)-P(A\cap B)
Example: Ace or King from a deck: P(\text{Ace})=\frac{4}{52}=\frac{1}{13},\; P(\text{King})=\frac{4}{52}=\frac{1}{13},\; P(\text{Ace}\cap \text{King})=0 \Rightarrow P(A\cup K)=\frac{2}{13}
Example: Ace or red card: P(A)=\tfrac{4}{52}=\tfrac{1}{13},\; P(\text{red})=\tfrac{26}{52}=\tfrac{1}{2},\; P(\text{Ace}\cap \text{red})=\tfrac{2}{52}=\tfrac{1}{26}
P(A\cup \text{red})=\tfrac{1}{13}+\tfrac{1}{2}-\tfrac{1}{26}=\tfrac{7}{13}

At least one success in n trials: P(\text{at least one})=1-P(\text{none})
No six in a single roll: P(\text{no six})=\tfrac{5}{6}
Two rolls: P(\text{at least one six})=1-(\tfrac{5}{6})^2=\tfrac{11}{36}
Seven rolls: P(\text{at least one six})=1-(\tfrac{5}{6})^7\approx 0.72
Four rolls for ace: P(\text{at least one ace})=1-(\tfrac{5}{6})^4
Twenty-four rolls for a double ace: P(\text{at least one double ace})=1-\left(\tfrac{35}{36}\right)^{24}\approx 0.491

Multiplication rule: P(A\cap B)=P(A)P(B|A)
If A and B are independent: P(A\cap B)=P(A)P(B)
For dependent events, use conditional probability: e.g., two-draw ace probability again: \frac{4}{52}\cdot\frac{3}{51}

Birthday problem threshold for probability > 0.5: about 256 people (ignoring leap years)
Computer reliability example: If each has success prob. 0.9 and independence holds, for 3:
0.9^3=0.729
Survival across multiple trials: If one-trial survival is 0.95, then 20 trials: 0.95^{20}\approx 0.36
Coupon/Spotify-type all-different problem (sampling without replacement approximation):
- If there are 100 artists and you draw n songs, the probability all are from different artists is
  \prod_{i=0}^{n-1} \left(1-\frac{i}{100}\right)
- For n=8: ≈ 0.750; for n=12: ≈ 0.50; for n=200: probability some artist not appear ≈ 0.134

Independent: P(A∩B)=P(A)P(B)
Mutually exclusive: P(A∪B)=P(A) + P(B) (and P(A∩B)=0)
Non-mutually exclusive: use P(A∪B)=P(A)+P(B)-P(A∩B)
When to multiply vs add
- Multiply: for P(A∩B) with independence or with conditional probability P(B|A)
- Add: for P(A∪B) when events can occur together, use subtraction to correct double-counting