🎲 Counting & Probability — Problem Types

Master the common problem patterns and systematic solution approaches for counting and probability problems.

Basic Counting Problems

Arrangement Problems

Recognition: Questions asking for number of ways to arrange objects Template:

1. Identify if order matters
2. Check for restrictions
3. Apply appropriate formula
4. Check answer

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

1. Order matters: Yes (different arrangements)
2. Restrictions: None
3. Formula: 5! = 120
4. Answer: 120 ways

Common variations:

• Linear arrangements
• Circular arrangements
• Arrangements with restrictions
• Arrangements with repetition

Selection Problems

Recognition: Questions asking for number of ways to choose objects Template:

1. Identify if order matters
2. Check for repetition
3. Apply appropriate formula
4. Check answer

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

1. Order matters: No (same group)
2. Repetition: No
3. Formula: C(5, 3) = 10
4. Answer: 10 ways

Common variations:

• Combinations
• Combinations with repetition
• Selections with restrictions
• Selections with conditions

Probability Problems

Basic Probability

Recognition: Questions asking for probability of an event Template:

1. Identify favorable outcomes
2. Identify total outcomes
3. Calculate probability
4. Check answer

Example: What is the probability of rolling a 3 on a fair die?

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

Common variations:

• Single event probability
• Complementary probability
• Probability with restrictions
• Probability with conditions

Compound Probability

Recognition: Questions asking for probability of multiple events Template:

1. Identify if events are independent
2. Apply appropriate formula
3. Calculate probability
4. Check answer

Example: What is the probability of rolling a 3 and then a 4 on two fair dice?

1. Independent: Yes (first roll doesn’t affect second)
2. Formula: P(3) × P(4) = (1/6) × (1/6) = 1/36
3. Answer: 1/36

Common variations:

• Independent events
• Dependent events
• Mutually exclusive events
• Overlapping events

Word Problems

Arrangement Word Problems

Recognition: Real-world situations involving arrangements Template:

1. Translate words to arrangement problem
2. Identify restrictions
3. Apply appropriate formula
4. Check answer

Example: How many ways can 5 people sit in a row if 2 specific people must sit together?

1. Translation: Linear arrangement with restriction
2. Restriction: 2 people must sit together
3. Formula: Treat 2 people as one unit: 4! × 2! = 48
4. Answer: 48 ways

Common variations:

• Seating arrangements
• Queue arrangements
• Team arrangements
• Prize arrangements

Selection Word Problems

Recognition: Real-world situations involving selections Template:

1. Translate words to selection problem
2. Identify restrictions
3. Apply appropriate formula
4. Check answer

Example: How many ways can 3 people be chosen from 5 if 2 specific people cannot both be chosen?

1. Translation: Combination with restriction
2. Restriction: 2 people cannot both be chosen
3. Formula: Total - Both chosen = C(5, 3) - C(3, 1) = 10 - 3 = 7
4. Answer: 7 ways

Common variations:

• Committee selections
• Team selections
• Prize selections
• Sample selections

Distribution Problems

Identical Objects to Distinct Boxes

Recognition: Questions about distributing identical objects Template:

1. Identify number of objects and boxes
2. Apply stars and bars formula
3. Calculate result
4. Check answer

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

1. Objects: 3 identical balls
2. Boxes: 5 distinct boxes
3. Formula: C(5 + 3 - 1, 3) = C(7, 3) = 35
4. Answer: 35 ways

Common variations:

• Identical objects to distinct boxes
• Identical objects to identical boxes
• Distinct objects to distinct boxes
• Distinct objects to identical boxes

Distinct Objects to Distinct Boxes

Recognition: Questions about distributing distinct objects Template:

1. Identify number of objects and boxes
2. Apply multiplication principle
3. Calculate result
4. Check answer

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

1. Objects: 3 distinct balls
2. Boxes: 5 distinct boxes
3. Formula: 5³ = 125
4. Answer: 125 ways

Common variations:

• Distinct objects to distinct boxes
• Distinct objects to identical boxes
• Identical objects to distinct boxes
• Identical objects to identical boxes

Probability Distribution Problems

Binomial Distribution

Recognition: Questions about number of successes in n trials Template:

1. Identify n, p, and k
2. Apply binomial formula
3. Calculate probability
4. Check answer

Example: What is the probability of getting exactly 3 heads in 5 coin flips?

1. n: 5 trials
2. p: 1/2 probability of success
3. k: 3 successes
4. Formula: C(5, 3) × (1/2)³ × (1/2)² = 10 × (1/8) × (1/4) = 5/16
5. Answer: 5/16

Common variations:

• Exact number of successes
• At least k successes
• At most k successes
• Range of successes

Geometric Distribution

Recognition: Questions about number of trials until first success Template:

1. Identify p and k
2. Apply geometric formula
3. Calculate probability
4. Check answer

Example: What is the probability of getting the first head on the 4th flip?

1. p: 1/2 probability of success
2. k: 4 trials
3. Formula: (1/2)³ × (1/2) = 1/16
4. Answer: 1/16

Common variations:

• First success on kth trial
• First success within k trials
• First success after k trials
• Expected number of trials

Conditional Probability Problems

Basic Conditional Probability

Recognition: Questions about probability given a condition Template:

1. Identify the condition
2. Apply conditional probability formula
3. Calculate probability
4. Check answer

Example: What is the probability of drawing a red card given that it’s a heart?

1. Condition: It’s a heart
2. Formula: P(red|heart) = P(red and heart) / P(heart)
3. Calculation: (13/52) / (13/52) = 1
4. Answer: 1

Common variations:

• Probability given condition
• Probability with restrictions
• Probability with information
• Probability with constraints

Bayes’ Theorem

Recognition: Questions about probability given test results Template:

1. Identify prior probabilities
2. Identify likelihood probabilities
3. Apply Bayes’ theorem
4. Calculate probability
5. Check answer

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?

1. Prior: P(disease) = 0.01
2. Likelihood: P(positive|disease) = 0.95, P(positive|no disease) = 0.02
3. Formula: P(disease|positive) = P(positive|disease) × P(disease) / P(positive)
4. Calculation: 0.95 × 0.01 / (0.95 × 0.01 + 0.02 × 0.99) ≈ 0.324
5. Answer: 0.324

Common variations:

• Medical testing
• Quality control
• Spam filtering
• Risk assessment

Expected Value Problems

Basic Expected Value

Recognition: Questions about average value Template:

1. Identify all possible values
2. Identify probabilities
3. Apply expected value formula
4. Calculate result
5. Check answer

Example: What is the expected value of rolling a fair die?

1. Values: 1, 2, 3, 4, 5, 6
2. Probabilities: 1/6 for each
3. Formula: E(X) = Σ(x × P(x))
4. Calculation: 1 × (1/6) + 2 × (1/6) + … + 6 × (1/6) = 3.5
5. Answer: 3.5

Common variations:

• Single random variable
• Multiple random variables
• Expected value of functions
• Expected value with conditions

Expected Value Applications

Recognition: Questions about decision making Template:

1. Identify all possible outcomes
2. Identify values and probabilities
3. Calculate expected value
4. Make decision
5. Check answer

Example: A game costs $5 to play. You roll a die and win $10 if you roll a 6, otherwise you win nothing. Should you play?

1. Outcomes: Roll 6 (win $10), Roll 1-5 (win $0)
2. Probabilities: 1/6 for 6, 5/6 for 1-5
3. Expected value: (1/6) × $10 + (5/6) × $0 = $1.67
4. Decision: Expected value ($1.67) < Cost ($5), so don’t play
5. Answer: No, don’t play

Common variations:

• Game theory
• Investment decisions
• Insurance decisions
• Risk assessment

Common Mistakes and Fixes

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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