🎲 Counting & Probability — Topics

Master the core topics and techniques for working with counting and probability in MATHCOUNTS.

Basic Counting Principles

Fundamental Counting Principle

Method: Multiply the number of ways to do each step Example: If there are 3 shirts and 4 pants, there are 3 × 4 = 12 ways to choose a shirt and pants

Pitfall: Forgetting to multiply all steps Fix: Always multiply the number of ways for each step

Addition Principle

Method: Add the number of ways to do mutually exclusive events Example: If there are 3 red shirts and 4 blue shirts, there are 3 + 4 = 7 ways to choose a shirt

Pitfall: Adding when events are not mutually exclusive Fix: Only add when events cannot happen together

Multiplication Principle

Method: Multiply the number of ways to do each step Example: If there are 3 ways to choose a shirt and 4 ways to choose pants, there are 3 × 4 = 12 ways to choose both

Pitfall: Forgetting to consider all steps Fix: Always consider all steps in the process

Permutations

Basic Permutations

Method: Use P(n, r) = n! / (n - r)! Example: P(5, 3) = 5! / (5 - 3)! = 5! / 2! = 120 / 2 = 60

Pitfall: Forgetting to subtract r from n Fix: Always use (n - r) in the denominator

Permutations with Repetition

Method: Use n^r Example: How many 3-digit numbers can be formed using digits 1-5 with repetition?

Answer: 5³ = 125

Pitfall: Using factorial instead of exponentiation Fix: Use exponentiation when repetition is allowed

Permutations with Restrictions

Method: Use multiplication principle and adjust for restrictions Example: How many ways can 5 people sit in a row if 2 specific people must sit together?

Treat the 2 people as one unit: 4 units
Arrange 4 units: 4! = 24 ways
Arrange the 2 people within their unit: 2! = 2 ways
Total: 24 × 2 = 48 ways

Pitfall: Not considering all restrictions Fix: Always consider all restrictions and adjust accordingly

Combinations

Basic Combinations

Method: Use C(n, r) = n! / (r!(n - r)!) Example: C(5, 3) = 5! / (3!2!) = 120 / (6 × 2) = 10

Pitfall: Forgetting to divide by r! and (n - r)! Fix: Always divide by both r! and (n - r)!

Combinations with Repetition

Method: Use C(n + r - 1, r) Example: How many ways can 3 identical balls be placed in 5 boxes?

Answer: C(5 + 3 - 1, 3) = C(7, 3) = 35

Pitfall: Using wrong formula Fix: Use C(n + r - 1, r) for combinations with repetition

Combinations with Restrictions

Method: Use addition principle and adjust for restrictions Example: How many ways can 3 people be chosen from 5 if 2 specific people cannot both be chosen?

Total ways: C(5, 3) = 10
Ways with both specific people: C(3, 1) = 3 (choose 1 more from remaining 3)
Answer: 10 - 3 = 7 ways

Pitfall: Not considering all restrictions Fix: Always consider all restrictions and adjust accordingly

Probability

Basic Probability

Method: Use P(event) = Number of favorable outcomes / Total number of outcomes Example: What is the probability of rolling a 3 on a fair die?

Favorable outcomes: 1 (rolling a 3)
Total outcomes: 6 (rolling 1, 2, 3, 4, 5, or 6)
Probability: 1/6

Pitfall: Not counting all possible outcomes Fix: Always count all possible outcomes

Complementary Probability

Method: Use P(not A) = 1 - P(A) Example: What is the probability of not rolling a 3 on a fair die?

P(not 3) = 1 - P(3) = 1 - 1/6 = 5/6

Pitfall: Forgetting to subtract from 1 Fix: Always subtract from 1 for complementary probability

Independent Events

Method: Use P(A and B) = P(A) × P(B) Example: What is the probability of rolling a 3 and then a 4 on two fair dice?

P(3) = 1/6, P(4) = 1/6
P(3 and 4) = (1/6) × (1/6) = 1/36

Pitfall: Multiplying when events are not independent Fix: Only multiply when events are independent

Dependent Events

Method: Use P(A and B) = P(A) × P(B|A) Example: What is the probability of drawing two red cards from a deck without replacement?

P(first red) = 26/52 = 1/2
P(second red|first red) = 25/51
P(both red) = (1/2) × (25/51) = 25/102

Pitfall: Not adjusting for the first event Fix: Always adjust for the first event in dependent events

Conditional Probability

Basic Conditional Probability

Method: Use P(A|B) = P(A and B) / P(B) Example: What is the probability of drawing a red card given that it’s a heart?

P(red and heart) = 13/52 = 1/4
P(heart) = 13/52 = 1/4
P(red|heart) = (1/4) / (1/4) = 1

Pitfall: Forgetting to divide by P(B) Fix: Always divide by P(B) for conditional probability

Bayes’ Theorem

Method: Use P(A|B) = P(B|A) × P(A) / P(B) Example: In a medical test, 95% of people with the disease test positive, and 2% of people without the disease test positive. If 1% of the population has the disease, what is the probability that someone who tests positive actually has the disease?

P(positive|disease) = 0.95
P(positive|no disease) = 0.02
P(disease) = 0.01
P(positive) = 0.95 × 0.01 + 0.02 × 0.99 = 0.0293
P(disease|positive) = 0.95 × 0.01 / 0.0293 ≈ 0.324

Pitfall: Forgetting to calculate P(B) Fix: Always calculate P(B) using the law of total probability

Expected Value

Basic Expected Value

Method: Use E(X) = Σ(x × P(x)) Example: What is the expected value of rolling a fair die?

E(X) = 1 × (1/6) + 2 × (1/6) + 3 × (1/6) + 4 × (1/6) + 5 × (1/6) + 6 × (1/6) = 3.5

Pitfall: Not considering all possible values Fix: Always consider all possible values of the random variable

Properties of Expected Value

Linearity: E(aX + bY) = aE(X) + bE(Y) Independence: If X and Y are independent, then E(XY) = E(X)E(Y)

Example: If E(X) = 3 and E(Y) = 4, then E(2X + 3Y) = 2(3) + 3(4) = 18

Pitfall: Forgetting linearity properties Fix: Always use linearity properties when applicable

Common Counting Problems

Arrangements

Linear arrangements: n! ways to arrange n distinct objects Circular arrangements: (n - 1)! ways to arrange n distinct objects in a circle Arrangements with restrictions: Use multiplication principle

Example: How many ways can 5 people sit in a circle?

Answer: (5 - 1)! = 4! = 24 ways

Pitfall: Using n! for circular arrangements Fix: Use (n - 1)! for circular arrangements

Selections

Combinations: C(n, r) ways to choose r objects from n Combinations with repetition: C(n + r - 1, r) ways to choose r objects from n with repetition Selections with restrictions: Use addition principle

Example: How many ways can 3 people be chosen from 5?

Answer: C(5, 3) = 10 ways

Pitfall: Using permutations instead of combinations Fix: Use combinations when order doesn’t matter

Distributions

Distinct objects to distinct boxes: n^r ways to distribute r distinct objects to n distinct boxes Identical objects to distinct boxes: C(n + r - 1, r) ways to distribute r identical objects to n distinct boxes Distinct objects to identical boxes: Use Stirling numbers (advanced)

Example: How many ways can 3 identical balls be placed in 5 boxes?

Answer: C(5 + 3 - 1, 3) = C(7, 3) = 35 ways

Pitfall: Using wrong formula for distribution type Fix: Always identify the type of distribution first

Common Probability Distributions

Binomial Distribution

Method: Use P(X = k) = C(n, k) × p^k × (1 - p)^(n - k) Example: What is the probability of getting exactly 3 heads in 5 coin flips?

P(X = 3) = C(5, 3) × (1/2)³ × (1/2)² = 10 × (1/8) × (1/4) = 10/32 = 5/16

Pitfall: Forgetting to include (1 - p)^(n - k) Fix: Always include both p^k and (1 - p)^(n - k)

Geometric Distribution

Method: Use P(X = k) = (1 - p)^(k - 1) × p Example: What is the probability of getting the first head on the 4th flip?

P(X = 4) = (1/2)³ × (1/2) = 1/16

Pitfall: Using k instead of (k - 1) Fix: Always use (k - 1) for geometric distribution

Hypergeometric Distribution

Method: Use P(X = k) = C(K, k) × C(N - K, n - k) / C(N, n) Example: What is the probability of drawing exactly 2 red cards from a deck of 52 cards in 5 draws?

P(X = 2) = C(26, 2) × C(26, 3) / C(52, 5) ≈ 0.325

Pitfall: Forgetting to adjust for sampling without replacement Fix: Always use hypergeometric distribution for sampling without replacement

Common Mistakes

Counting Mistakes

Mistake: Using permutations instead of combinations Fix: Use permutations when order matters, combinations when it doesn’t

Mistake: Forgetting to consider restrictions Fix: Always consider all restrictions and adjust accordingly

Mistake: Using wrong formula for repetition Fix: Use n^r for permutations with repetition, C(n + r - 1, r) for combinations with repetition

Probability Mistakes

Mistake: Not counting all possible outcomes Fix: Always count all possible outcomes

Mistake: Multiplying when events are not independent Fix: Only multiply when events are independent

Mistake: Forgetting to adjust for dependent events Fix: Always adjust for the first event in dependent events

Expected Value Mistakes

Mistake: Not considering all possible values Fix: Always consider all possible values of the random variable

Mistake: Forgetting linearity properties Fix: Always use linearity properties when applicable

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🎲 Counting & Probability — Topics#

Basic Counting Principles#

Fundamental Counting Principle#

Addition Principle#

Multiplication Principle#

Permutations#

Basic Permutations#

Permutations with Repetition#

Permutations with Restrictions#

Combinations#

Basic Combinations#

Combinations with Repetition#

Combinations with Restrictions#

Probability#

Basic Probability#

Complementary Probability#

Independent Events#

Dependent Events#

Conditional Probability#

Basic Conditional Probability#

Bayes’ Theorem#

Expected Value#

Basic Expected Value#

Properties of Expected Value#

Common Counting Problems#

Arrangements#

Selections#

Distributions#

Common Probability Distributions#

Binomial Distribution#

Geometric Distribution#

Hypergeometric Distribution#

Common Mistakes#

Counting Mistakes#

Probability Mistakes#

Expected Value Mistakes#