Problem
Problem
📋 Step 1 — Identify the ingredients. Fill in the values from the problem:
🧮 Step 2 — Now apply Bayes and compute the final answer:
P(H|E) = P(E|H) × P(H) / [ P(E|H)×P(H) + P(E|¬H)×P(¬H) ]
Problem
📋 Step 1 — Fill in the ingredients and the three calculation pieces:
🧮 Step 2 — Multiply the three pieces together for the final answer:
P(X=k) = C(n,k) × p^k × (1−p)^(n−k)
Problem
📋 Step 1 — Identify the four ingredients:
🧮 Step 2 — Compute the three combinations, then divide:
P(X=k) = C(K, k) × C(N−K, n−k) / C(N, n)
Problem
📋 Step 1 — Find P(none) first:
🧮 Step 2 — Now compute 1 − P(none):
P(at least 1) = 1 − q^n
Problem
📋 Step 1 — Fill in the probability distribution. All P(x) must sum to 1:
🧮 Step 2 — Compute each x · P(x), then sum them for E[X]:
E[X] = x₁·P(x₁) + x₂·P(x₂) + … + xₙ·P(xₙ)
Problem
Problem
Total
Multiply all slot counts together
Problem
Problem
Problem
C(n+k−1, k−1)
Final answer — number of distributions
Problem
Which tool does this problem call for?
Conditional Probability
Formula: P(A|B) = P(A ∩ B) / P(B)
Read as: "Probability of A given B"
When to use: When a condition shrinks your sample space
Key signal words: "given that", "if", "of those who"
Bayes Theorem
Formula: P(H|E) = P(E|H) × P(H) / P(E)
P(H): Prior — base rate before evidence
P(E|H): Likelihood — prob of evidence if H is true
P(E): Normalizer — total prob of evidence
P(E): = P(E|H)×P(H) + P(E|¬H)×P(¬H)
Key signal words: "what's the probability it came from / was caused by"
Binomial Distribution
Formula: P(X=k) = C(n,k) × p^k × (1-p)^(n-k)
n: Number of trials
k: Number of successes you want
p: Probability of success per trial
C(n,k): n! / (k! × (n-k)!)
Key signal words: "exactly k times", "out of n trials", "independent"
Hypergeometric Distribution
Formula: P(X=k) = C(K,k) × C(N−K, n−k) / C(N,n)
N: Total population size
K: Number of successes in the population
n: Number of draws (sample size)
k: Successes you want to see in your draw
Key signal: "WITHOUT replacement" — pool shrinks each draw
vs Binomial: Binomial = WITH replacement (p stays constant each trial)
Complement Rule — "At Least"
Formula: P(at least 1) = 1 − P(none) = 1 − q^n
q: Probability of failure per trial = 1 − p
Why: Counting "at least one" directly is tedious — count zero, then subtract
Key signal words: "at least one", "at least once", "one or more"
Expected Value E[X]
Formula: E[X] = Σ x · P(x)
Read as: Multiply each outcome by its probability, then add them all up
Important: E[X] is the long-run average — NOT the most likely outcome
Binomial shortcut: E[X] = n × p
Multiplication Principle
Formula: Total = n₁ × n₂ × … × nₖ
When to use: Independent sequential choices — each stage multiplies the total
Key signal words: "how many ways", "choices", "combinations" (informal), "sequence"
Constraint tip: Restrictions (no leading zero, no repeats) just change one slot's count
Permutations
Standard P(n,r): P(n,r) = n! / (n−r)!
n: Total items in the pool
r: Positions to fill (ORDER matters)
Multinomial: n! / (n₁! × n₂! × … × nₖ!)
Multinomial use: Arranging objects where some are identical (e.g. letters in a word)
vs Combinations: Permutations = order matters; C(n,r) = order doesn't matter
Key signal words: "arranged", "ordered", "assigned to distinct roles", "first/second/third place"
Inclusion-Exclusion
2-set: |A∪B| = |A| + |B| − |A∩B|
3-set: |A∪B∪C| = |A|+|B|+|C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|
Why: Adding sets double-counts overlaps; subtract them. With 3 sets the triple overlap is over-subtracted — add it back once.
Key signal words: "at least one of", "in A or B", "union", "how many total"
Stars & Bars
Formula: C(n+k−1, k−1)
n: Identical objects to distribute
k: Distinct bins (recipients)
Assumes: Each bin can receive 0 or more objects
Lower bound trick: If each bin needs ≥ m, substitute n′ = n − k·m first, then apply formula
Key signal words: "distribute", "identical", "non-negative integer solutions to x₁+…+xₖ=n"
Quick Decision Guide
→ "At least once / at least one"? COMPLEMENT = 1 − q^n
→ Repeated trials WITH replacement, "exactly k"? BINOMIAL
→ Drawing WITHOUT replacement, finite pool? HYPERGEOMETRIC
→ "Given that / if" reduces sample space? CONDITIONAL
→ Flipping cause↔effect, updating belief? BAYES
→ Binomial likelihood then Bayes update? BAYES + BINOMIAL
→ Average outcome over many trials? EXPECTED VALUE
→ Independent sequential choices, count arrangements? MULTIPLICATION PRINCIPLE
→ Ordered arrangement from a pool (order matters)? PERMUTATIONS P(n,r)
→ Repeated letters / identical items, arrangements? MULTINOMIAL n!/(n₁!×…)
→ Count elements in union of overlapping sets? INCLUSION-EXCLUSION
→ Distribute identical objects into distinct bins? STARS & BARS C(n+k−1,k−1)