Core Concepts & Formulas
Topic 1
Sample Space & Basic Probability
P(A) = (Number of favorable outcomes) / (Total outcomes in S)
0 ≤ P(A) ≤ 1 | P(S) = 1 | P(∅) = 0
Sample space S: set of all possible outcomes of an experiment.
Event A is any subset of S. Complement: P(A') = 1 - P(A).
P(A') = 1 - P(A)
P(S) = 1
Topic 2
Addition Rule (Union)
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
If A and B are mutually exclusive: P(A ∪ B) = P(A) + P(B)
Mutually exclusive means A ∩ B = ∅, so they cannot both occur.
Union = sum - overlap
Topic 3
Conditional Probability
P(A | B) = P(A ∩ B) / P(B), where P(B) > 0
P(A | B): probability of A given that B has already occurred.
Rearranged: P(A ∩ B) = P(B) · P(A | B)
Condition narrows sample space to B
Topic 4
Independence
A ⊥ B iff P(A ∩ B) = P(A) · P(B)
Equivalently: P(A | B) = P(A)
Independence means knowing B gives no information about A.
Mutually exclusive ≠ Independent (unless P(A) = 0 or P(B) = 0).
Independent → multiply probabilities
Topic 5
Permutations & Combinations
Permutation (order matters): P(n,r) = n! / (n-r)!
Combination (order doesn't): C(n,r) = n! / [r!(n-r)!]
Circular permutation: (n-1)!
Use P when arrangement/sequence matters; C when only selection matters.
C(n,r) = C(n, n-r)
Circular = (n-1)!
Topic 6
Total Probability & Bayes' Theorem
Total: P(B) = Σ P(B|Aᵢ)·P(Aᵢ)
Bayes: P(Aᵢ|B) = P(B|Aᵢ)·P(Aᵢ) / P(B)
Partition {A₁,…,Aₙ}: mutually exclusive, collectively exhaustive events.
Bayes: reverse conditional
Topic 7
Binomial Probability
P(X = k) = C(n,k) · pᵏ · (1-p)^(n-k)
E[X] = np | Var(X) = np(1-p)
n = trials, k = successes, p = probability of success per trial.
Trials must be independent with constant p.
C(n,k) · pᵏ · qⁿ⁻ᵏ
Worked Examples
Example A · Conditional Probability
A box has 4 red and 6 blue balls. Two are drawn without replacement. Find P(2nd is red | 1st is red).
After drawing 1 red, 3 red remain out of 9 total.
P(2nd red | 1st red) = 3/9 = 1/3
Answer: 1/3
Example B · Combination
How many ways can a committee of 3 be chosen from 7 people?
C(7,3) = 7! / (3! · 4!) = (7×6×5) / (3×2×1) = 210 / 6 = 35
Answer: 35
Example C · Binomial
A fair coin is flipped 4 times. Find P(exactly 3 heads).
P(X=3) = C(4,3)·(1/2)³·(1/2)¹ = 4 · 1/8 · 1/2 = 4/16 = 1/4
Answer: 1/4