Chance and Probability: Investment Banking Brain Teasers

Investment banking interviews often feature probability and chance-based brain teasers to test analytical thinking, problem-solving skills, and comfort with uncertainty. These questions assess your ability to think clearly under pressure while applying mathematical reasoning to complex scenarios.

10 Core Brain Teasers covering:

• Classic probability problems (cards, dice)

• Market scenarios (stock picking, IPO allocation)

• Trading floor situations (profit probability, market making)

• Logic puzzles (analyst reasoning, merger probability)

• Advanced concepts (hedge fund returns, portfolio optimization)

Key Features:

• Step-by-step solutions for each problem

• Real-world financial context connecting math to business

• Interview insights explaining what each question tests

• Common mistakes and how to avoid them

• Strategic tips for approaching probability problems in interviews

Interview Preparation Value:

• Tests analytical thinking under pressure

• Demonstrates quantitative skills essential for banking roles

• Shows ability to connect mathematical concepts to business decisions

• Practices clear communication of complex reasoning

The problems range from fundamental probability concepts to sophisticated scenarios involving Bayes' theorem, game theory, and portfolio optimization. Each solution includes both the mathematical approach and the business intuition, preparing candidates for the analytical challenges they'll face in investment banking interviews and on the job.

This collection serves as both a study guide for interview preparation and a reference for the types of probabilistic thinking that drive success in quantitative finance roles.

Classic Card and Dice Problems

Brain Teaser 1: The Ace Problem

Question: You draw cards one by one from a standard 52-card deck without replacement. What is the probability that the first ace appears on the 5th draw?

Solution: Step 1: Set up the problem

• We need exactly 4 non-aces followed by 1 ace

• First 4 cards must be non-aces, 5th card must be an ace

Step 2: Calculate probabilities for each draw

• P(1st card is not ace) = 48/52

• P(2nd card is not ace | 1st not ace) = 47/51

• P(3rd card is not ace | first 2 not aces) = 46/50

• P(4th card is not ace | first 3 not aces) = 45/49

• P(5th card is ace | first 4 not aces) = 4/48

Step 3: Multiply all probabilities P = (48/52) × (47/51) × (46/50) × (45/49) × (4/48) = 0.0299 or 2.99%

Interview Insight: This tests understanding of conditional probability and systematic thinking.

Brain Teaser 2: The Dice Paradox

Question: You roll two fair six-sided dice. Given that at least one die shows a 6, what is the probability that both dice show 6?

Solution: Step 1: Identify the given condition

• Event A: Both dice show 6

• Event B: At least one die shows 6

• We need P(A|B)

Step 2: Count favorable outcomes

• Total outcomes with at least one 6: 11 outcomes

  • (6,1), (6,2), (6,3), (6,4), (6,5), (6,6), (1,6), (2,6), (3,6), (4,6), (5,6)

• Outcomes with both 6s: 1 outcome (6,6)

Step 3: Apply conditional probability P(A|B) = 1/11 ≈ 9.09%

Common Mistake: Many assume it's 1/6, forgetting the conditioning effect.

Market-Based Scenarios

Brain Teaser 3: The Stock Picker's Dilemma

Question: An analyst claims to pick winning stocks 70% of the time. You observe her next 10 picks, and 8 are winners. What's the probability she's truly skilled (vs. lucky) if you initially believed there was a 20% chance she had real skill?

Solution: Step 1: Set up Bayes' Theorem

• Prior: P(Skilled) = 0.2, P(Lucky) = 0.8

• Likelihood: P(8/10 wins | Skilled) vs P(8/10 wins | Lucky)

Step 2: Calculate likelihoods

• If skilled (70% win rate): P(8/10) = C(10,8) × (0.7)^8 × (0.3)^2 = 0.233

• If lucky (50% win rate): P(8/10) = C(10,8) × (0.5)^10 = 0.044

Step 3: Apply Bayes' Theorem

• P(Skilled | 8/10) = [0.233 × 0.2] / [0.233 × 0.2 + 0.044 × 0.8]

• P(Skilled | 8/10) = 0.0466 / (0.0466 + 0.0352) = 0.57 or 57%

Investment Banking Relevance: This mirrors real decisions about fund manager selection and due diligence.

Brain Teaser 4: The IPO Allocation Game

Question: You're allocating shares in a hot IPO to 100 clients. Each client wants 1000 shares, but you only have 50,000 shares available. If you randomly select 50 clients to receive full allocations, what's the probability that your top 10 VIP clients all get shares?

Solution: Step 1: Frame as a hypergeometric problem

• Population: 100 clients

• Success states: 10 VIP clients

• Sample size: 50 selected clients

• We want all 10 VIPs selected

Step 2: Calculate using hypergeometric distribution P = C(10,10) × C(90,40) / C(100,50)

Step 3: Compute the probability

• This equals C(90,40) / C(100,50)

• Using combinations: ≈ 0.000006 or 0.0006%

Alternative thinking: Probability = (50/100) × (49/99) × ... × (41/91) ≈ 0.000006%

Business Insight: Shows why strategic allocation (not random) is typically used in practice.

Trading Floor Scenarios

Brain Teaser 5: The Trader's Coin Flip

Question: A trader makes money on 60% of trades. Each winning trade makes $1000, each losing trade loses $800. After 100 trades, what's the probability of being profitable overall?

Solution: Step 1: Calculate expected profit per trade

• E[Profit] = 0.6 × $1000 + 0.4 × (-$800) = $600 - $320 = $280

Step 2: Calculate variance per trade

• E[Profit²] = 0.6 × (1000)² + 0.4 × (800)² = 600,000 + 256,000 = 856,000

• Var = 856,000 - (280)² = 856,000 - 78,400 = 777,600

• Standard deviation per trade = √777,600 = $882

Step 3: Apply Central Limit Theorem for 100 trades

• Total expected profit = 100 × $280 = $28,000

• Standard deviation of total = √100 × $882 = $8,820

• Z-score for breaking even: Z = (0 - 28,000) / 8,820 = -3.17

Step 4: Find probability P(Profitable) = P(Z > -3.17) ≈ 99.92%

Trading Insight: Even with positive expected value, understanding the distribution of outcomes is crucial for risk management.

Brain Teaser 6: The Options Market Maker

Question: You're a market maker for options. On any given trade, you have a 52% chance of making $100 and a 48% chance of losing $100. You execute 10,000 trades per year. What's the probability you'll lose money in a year?

Solution: Step 1: Calculate per-trade statistics

• Expected profit per trade = 0.52 × $100 + 0.48 × (-$100) = $4

• Variance per trade = 0.52 × (100)² + 0.48 × (100)² - (4)² = 10,000 - 16 = 9,984

• Standard deviation per trade = $99.92 ≈ $100

Step 2: Calculate annual statistics

• Expected annual profit = 10,000 × $4 = $40,000

• Annual standard deviation = √10,000 × $100 = $10,000

Step 3: Find probability of loss

• Z-score for breaking even: Z = (0 - 40,000) / 10,000 = -4

• P(Loss) = P(Z < -4) ≈ 0.00003 or 0.003%

Market Making Insight: High-frequency trading with small edges can be very profitable with sufficient volume.

Logic and Game Theory

Brain Teaser 7: The Analyst's Dilemma

Question: Three analysts are in a room. Each has either a red or blue hat, determined by coin flips. Each can see the others' hats but not their own. They're told that at least one hat is red. If an analyst can deduce their hat color, they should raise their hand. All three raise their hands simultaneously. What color hats do they have?

Solution: Step 1: Analyze the logical constraints

• At least one hat is red (given)

• All three raised hands simultaneously

• Each can see the other two hats

Step 2: Consider possible scenarios

• If someone saw two blue hats, they'd know theirs was red (to satisfy "at least one red")

• If this happened, only one person would raise their hand

• Since all three raised hands, no one saw two blue hats

Step 3: Deduce the answer

• Each person sees at least one red hat on others

• For all three to deduce simultaneously, each must see exactly one red hat

• This is only possible if all three hats are red

Answer: All three analysts have red hats.

Interview Value: Tests logical reasoning and ability to think from multiple perspectives.

Brain Teaser 8: The Merger Probability

Question: Company A is trying to acquire Company B. The deal has a 60% chance of regulatory approval. If approved, there's an 80% chance Company C will make a competing bid. If Company C bids, there's a 70% chance they'll win. What's the probability Company A successfully acquires Company B?

Solution: Step 1: Map out the probability tree

• P(Regulatory approval) = 0.6

• P(No approval) = 0.4

• P(Company C bids | Approved) = 0.8

• P(Company C wins | C bids) = 0.7

Step 2: Calculate success scenarios for Company A Company A succeeds if:

1. No regulatory approval: P = 0 (deal fails)

2. Approval AND no competing bid: P = 0.6 × 0.2 = 0.12

3. Approval AND competing bid AND Company A wins: P = 0.6 × 0.8 × 0.3 = 0.144

Step 3: Sum the probabilities P(Company A succeeds) = 0 + 0.12 + 0.144 = 0.264 or 26.4%

M&A Insight: Real deal probabilities involve complex interdependent factors that require careful modeling.

Advanced Probability Concepts

Brain Teaser 9: The Hedge Fund Returns

Question: A hedge fund claims their monthly returns are independent and normally distributed with mean 2% and standard deviation 5%. What's the probability they'll have a positive return every month for the next year?

Solution: Step 1: Calculate monthly probability of positive return

• Monthly return ~ N(0.02, 0.05²)

• P(Monthly return > 0) = P(Z > (0-0.02)/0.05) = P(Z > -0.4) = 0.6554

Step 2: Calculate probability for all 12 months

• Since returns are independent: P(All 12 positive) = (0.6554)^12 = 0.0009 or 0.09%

Step 3: Consider the business implication

• Extremely unlikely under stated assumptions

• Suggests either returns aren't independent, distribution parameters are wrong, or fund is very skilled/lucky

Due Diligence Insight: Such claims should trigger deeper investigation into the fund's strategy and risk management.

Brain Teaser 10: The Portfolio Rebalancing Puzzle

Question: You have two assets that are perfectly negatively correlated. Asset A has expected return 10% with volatility 20%. Asset B has expected return 6% with volatility 30%. What portfolio weights minimize risk while achieving at least 8% expected return?

Solution: Step 1: Set up the optimization problem

• Let w = weight in Asset A, (1-w) = weight in Asset B

• Expected return constraint: 10w + 6(1-w) ≥ 8

• This gives us: 4w ≥ 2, so w ≥ 0.5

Step 2: Calculate portfolio volatility with perfect negative correlation

• σ²_portfolio = w²(0.2)² + (1-w)²(0.3)² + 2w(1-w)(0.2)(0.3)(-1)

• σ²_portfolio = 0.04w² + 0.09(1-w)² - 0.12w(1-w)

• σ²_portfolio = 0.04w² + 0.09(1-2w+w²) - 0.12w + 0.12w²

• σ²_portfolio = 0.25w² - 0.3w + 0.09

Step 3: Minimize variance subject to constraint

• Taking derivative: d(σ²)/dw = 0.5w - 0.3 = 0

• Optimal w = 0.6, but we need w ≥ 0.5 for return constraint

• Since 0.6 > 0.5, the unconstrained optimum satisfies our constraint

Answer: 60% in Asset A, 40% in Asset B

• Expected return: 0.6(10%) + 0.4(6%) = 8.4%

• Portfolio volatility: σ = √(0.25(0.6)² - 0.3(0.6) + 0.09) = 3%

Portfolio Theory Insight: Perfect negative correlation allows for significant risk reduction while maintaining target returns.

Interview Strategy Tips

How to Approach Probability Brain Teasers:

1. Listen Carefully: Make sure you understand all constraints and assumptions

2. Ask Clarifying Questions: Don't hesitate to verify ambiguous details

3. Think Out Loud: Walk through your reasoning step-by-step

4. Start Simple: Break complex problems into manageable pieces

5. Check Your Answer: Does the result make intuitive sense?

6. Consider Edge Cases: What happens at extremes?

7. Connect to Finance: Explain why this type of thinking matters in banking

Common Pitfalls to Avoid:

• Conditional Probability Confusion: Remember to properly condition on given information

• Independence Assumptions: Don't assume independence unless explicitly stated

• Base Rate Neglect: Consider prior probabilities in Bayesian problems

• Survivorship Bias: Account for unseen failures in performance analysis

• Sample Size Effects: Small samples can be very misleading

Advanced Techniques:

• Symmetry Arguments: Look for ways to simplify using symmetry

• Complement Thinking: Sometimes it's easier to calculate 1 - P(opposite)

• Recursive Approaches: Break problems into smaller, similar subproblems

• Approximation Methods: Know when normal approximations are valid

• Simulation Thinking: Consider how you'd test your answer with Monte Carlo methods

Conclusion

These brain teasers represent the type of analytical thinking valued in investment banking. They test not just mathematical ability, but also:

• Structured Problem Solving: Breaking complex scenarios into manageable steps

• Risk Assessment: Understanding uncertainty and its financial implications

• Logical Reasoning: Drawing correct conclusions from incomplete information

• Communication Skills: Explaining complex concepts clearly under pressure

• Business Intuition: Connecting mathematical results to real-world decisions

Success with these problems demonstrates the quantitative skills and clear thinking essential for roles in trading, risk management, corporate finance, and investment analysis. The key is practice, pattern recognition, and maintaining composure while working through unfamiliar scenarios.

Remember: In actual interviews, the process of solving the problem is often more important than getting the exact right answer. Show your thinking, explain your assumptions, and demonstrate how you approach uncertainty with mathematical rigor and business sense.