Probability and Logic: Investment Banking Puzzles

Understand about Probability and Logic

Investment banking interviews frequently test candidates' ability to think analytically under pressure through probability and logic puzzles. These questions assess quantitative reasoning, structured problem-solving, and the ability to make sound decisions with incomplete information all critical skills for success in finance.

10 Core Puzzles organized into strategic sections:

1. Fundamental Probability - Bayes' theorem applications in trading signals and capital allocation

2. Market Logic - Information asymmetry and merger arbitrage scenarios

3. Advanced Logic - Multi-person reasoning and consistency checking

4. Risk & Decision Theory - Incentive alignment and manager selection

5. Game Theory - Strategic thinking in IPOs and information cascades

Key Features:

• Step-by-step mathematical solutions with clear reasoning

• Business context connecting each puzzle to real banking situations

• Strategic insights explaining what skills each puzzle tests

• Interview tips for approaching similar problems under pressure

Skills Assessed:

• Bayesian probability and conditional reasoning

• Expected value calculations and decision theory

• Game theory and strategic thinking

• Logic puzzles requiring systematic analysis

• Risk assessment with incomplete information

Interview Preparation Value:

• Tests analytical thinking under time pressure

• Demonstrates quantitative skills essential for banking roles

• Shows ability to connect mathematical concepts to business decisions

• Practices clear communication of complex reasoning

The puzzles range from classic probability problems to sophisticated scenarios involving information asymmetry, incentive alignment, and strategic decision-making. Each solution includes both rigorous mathematical analysis and practical business insights, preparing candidates for the types of analytical challenges they'll face in investment banking interviews and daily work.

This collection serves as both a comprehensive study guide and a reference for the probabilistic and logical thinking that drives success in quantitative finance roles.

Section 1: Fundamental Probability Puzzles

Puzzle 1: The False Positive Trading Signal

Scenario: Your trading algorithm generates buy signals. Historical data shows that 5% of stocks actually outperform the market significantly. Your algorithm correctly identifies 90% of outperforming stocks but also gives false positives for 15% of non-outperforming stocks. If your algorithm flags a stock, what's the probability it will actually outperform?

Solution Process:

Step 1: Define the events

• P(Outperform) = 0.05

• P(No Outperform) = 0.95

• P(Signal | Outperform) = 0.90 (sensitivity)

• P(Signal | No Outperform) = 0.15 (false positive rate)

Step 2: Apply Bayes' Theorem We need P(Outperform | Signal)

P(Signal) = P(Signal | Outperform) × P(Outperform) + P(Signal | No Outperform) × P(No Outperform) P(Signal) = 0.90 × 0.05 + 0.15 × 0.95 = 0.045 + 0.1425 = 0.1875

Step 3: Calculate posterior probability P(Outperform | Signal) = P(Signal | Outperform) × P(Outperform) / P(Signal) P(Outperform | Signal) = (0.90 × 0.05) / 0.1875 = 0.045 / 0.1875 = 0.24 or 24%

Key Insight: Even with a 90% accurate algorithm, the low base rate of outperforming stocks means most signals are false positives. This demonstrates the importance of base rates in financial decision-making.

Puzzle 2: The Coin Flip Capital Allocation

Scenario: You manage a $100M portfolio. Each investment opportunity is like a coin flip: 50% chance of +20% return, 50% chance of -10% return. You can make any number of independent bets. How should you size each bet to maximize long-term growth?

Solution Process:

Step 1: Apply Kelly Criterion Kelly fraction = (bp - q) / b Where: b = odds on winning, p = win probability, q = loss probability

Step 2: Calculate parameters

• p = 0.5 (50% win probability)

• q = 0.5 (50% loss probability)

• When winning: gain 20% of bet, so b = 0.20/0.10 = 2 (relative to loss amount)

• Actually, let's recalculate more carefully:

Step 3: Reframe for Kelly

• Win probability: p = 0.5

• Win amount: +20% of bet

• Loss amount: -10% of bet

• Net odds: when you win, you gain 20%; when you lose, you lose 10%

Kelly fraction = (p × gain - q × loss) / gain Kelly fraction = (0.5 × 0.20 - 0.5 × 0.10) / 0.20 = (0.10 - 0.05) / 0.20 = 0.25

Answer: Allocate 25% of capital to each bet for optimal long-term growth.

Investment Insight: Even with positive expected value, over-betting can lead to ruin. The Kelly criterion balances growth and risk of total loss.

Section 2: Market Logic Puzzles

Puzzle 3: The Insider Trading Paradox

Scenario: In a market, there are three types of traders: 60% uninformed, 30% moderately informed, 10% highly informed. An uninformed trader makes money 45% of the time, a moderately informed trader 55% of the time, and a highly informed trader 80% of the time. You observe a trader make money on 7 out of 10 consecutive trades. What's the probability they're highly informed?

Solution Process:

Step 1: Set up prior probabilities

• P(Uninformed) = 0.60

• P(Moderately Informed) = 0.30

• P(Highly Informed) = 0.10

Step 2: Calculate likelihoods using binomial distribution P(7 wins out of 10 | Uninformed) = C(10,7) × (0.45)^7 × (0.55)^3 = 120 × 0.0037 × 0.166 = 0.074

P(7 wins out of 10 | Moderately Informed) = C(10,7) × (0.55)^7 × (0.45)^3 = 120 × 0.0152 × 0.091 = 0.166

P(7 wins out of 10 | Highly Informed) = C(10,7) × (0.80)^7 × (0.20)^3 = 120 × 0.2097 × 0.008 = 0.201

Step 3: Apply Bayes' Theorem P(Highly Informed | 7 wins) = [P(7 wins | Highly Informed) × P(Highly Informed)] / P(7 wins)

P(7 wins) = 0.074 × 0.60 + 0.166 × 0.30 + 0.201 × 0.10 = 0.0444 + 0.0498 + 0.0201 = 0.1143

P(Highly Informed | 7 wins) = (0.201 × 0.10) / 0.1143 = 0.0201 / 0.1143 = 0.176 or 17.6%

Market Insight: Even exceptional performance might not indicate insider information due to the low base rate of informed traders.

Puzzle 4: The Merger Arbitrage Decision Tree

Scenario: Company A announces intent to acquire Company B for $50/share. B currently trades at $45. The deal has a 70% chance of regulatory approval. If approved, there's a 20% chance of a competing bid at $55. If there's a competing bid, the original deal fails 60% of the time. What's the expected value of buying B's stock?

Solution Process:

Step 1: Map out all possible outcomes

1. No regulatory approval (30%): Stock returns to $40 (pre-announcement price)

2. Approval, no competing bid (70% × 80% = 56%): Deal closes at $50

3. Approval, competing bid, original deal wins (70% × 20% × 40% = 5.6%): Deal closes at $50

4. Approval, competing bid, original deal fails (70% × 20% × 60% = 8.4%): Competing bid wins at $55

Step 2: Calculate expected value E[Stock Price] = 0.30 × $40 + 0.56 × $50 + 0.056 × $50 + 0.084 × $55 E[Stock Price] = $12 + $28 + $2.80 + $4.62 = $47.42

Step 3: Determine trading decision Current price: $45 Expected value: $47.42 Expected profit: $47.42 - $45 = $2.42 per share

Decision: Buy the stock, as it offers positive expected value.

Arbitrage Insight: Merger arbitrage requires careful probability assessment of multiple regulatory and competitive scenarios.

Section 3: Advanced Logic Puzzles

Puzzle 5: The Three Analysts Problem

Scenario: Three analysts (A, B, C) are told that each has a number on their forehead—either 1, 2, or 3—and all three numbers are different. Each can see the others' numbers but not their own. They're told that at least one number is greater than 1. Each analyst, if they can deduce their number, should raise their hand. All three raise their hands simultaneously. What numbers do they have?

Solution Process:

Step 1: Analyze the constraint

• All numbers are different and from {1, 2, 3}

• At least one number > 1 (so not all 1s, which is impossible anyway)

• All three deduced their numbers simultaneously

Step 2: Consider what each analyst sees If any analyst saw two 1s on the others, they'd immediately know they have either 2 or 3 (since all must be different). But they couldn't distinguish between 2 and 3 without more information.

Step 3: Use simultaneity as key information For all three to deduce simultaneously, each must see information that uniquely determines their number. This happens only if each sees the numbers {1, 2} on the others.

Step 4: Verify the solution

• A sees {1, 2}, knows they must have 3

• B sees {1, 3}, knows they must have 2

• C sees {2, 3}, knows they must have 1

Answer: The three analysts have numbers 3, 2, and 1 respectively.

Logic Insight: Simultaneous deduction often indicates symmetric information conditions.

Puzzle 6: The Bond Trader's Dilemma

Scenario: A bond trader makes the following claim: "Either the Fed will raise rates next month, or the yield curve will invert, or both." You know that historically, when the Fed raises rates, the yield curve inverts 80% of the time. The Fed raises rates 30% of the time overall, and the yield curve inverts 25% of the time overall. Is the trader's statement logically consistent with these probabilities?

Solution Process:

Step 1: Define events and probabilities

• R: Fed raises rates, P(R) = 0.30

• I: Yield curve inverts, P(I) = 0.25

• P(I|R) = 0.80 (curve inverts given rate increase)

Step 2: Calculate P(I ∩ R) P(I ∩ R) = P(I|R) × P(R) = 0.80 × 0.30 = 0.24

Step 3: Check for consistency We need P(I ∪ R) using the inclusion-exclusion principle: P(I ∪ R) = P(I) + P(R) - P(I ∩ R) = 0.25 + 0.30 - 0.24 = 0.31

Step 4: Verify logical consistency The trader claims P(I ∪ R) = 1 (certainty that at least one occurs). But our calculation shows P(I ∪ R) = 0.31.

Answer: The trader's statement is NOT logically consistent with the historical probabilities.

Trading Insight: Market predictions must be consistent with underlying probability structures to be credible.

Section 4: Risk and Decision Theory

Puzzle 7: The Hedge Fund Manager's Choice

Scenario: A hedge fund manager can choose between two strategies:

• Strategy A: +10% with 60% probability, -5% with 40% probability

• Strategy B: +15% with 40% probability, -8% with 60% probability

The manager's bonus depends on positive returns only. If positive, bonus = 10% of returns. If negative, no bonus. Which strategy should a rational manager choose?

Solution Process:

Step 1: Calculate expected returns Strategy A: E[Return] = 0.60 × 10% + 0.40 × (-5%) = 6% - 2% = 4% Strategy B: E[Return] = 0.40 × 15% + 0.60 × (-8%) = 6% - 4.8% = 1.2%

Step 2: Calculate expected bonus Strategy A: E[Bonus] = 0.60 × (10% × 10%) + 0.40 × 0 = 0.60 × 1% = 0.6% Strategy B: E[Bonus] = 0.40 × (10% × 15%) + 0.60 × 0 = 0.40 × 1.5% = 0.6%

Step 3: Analyze the incentive alignment

• From fund's perspective: Strategy A is better (4% vs 1.2% expected return)

• From manager's perspective: Both strategies have equal expected bonus (0.6%)

• However, Strategy A has higher probability of positive return (60% vs 40%)

Answer: The rational manager should choose Strategy A it's better for the fund and has higher probability of bonus payment.

Agency Problem Insight: Even when expected bonuses are equal, incentive structures can still align with fund interests through probability considerations.

Puzzle 8: The Portfolio Manager's Paradox

Scenario: You're comparing two portfolio managers over 5 years:

• Manager X: Returns of +10%, +5%, -3%, +8%, +12% (beats benchmark in 4/5 years)

• Manager Y: Returns of +15%, -2%, +9%, +6%, +4% (beats benchmark in 4/5 years)

The benchmark returns were +8%, +3%, -5%, +6%, +10%. Both managers have the same Sharpe ratio. An investor says, "I prefer Manager X because they're more consistent." Is this preference rational?

Solution Process:

Step 1: Calculate total returns Manager X: (1.10)(1.05)(0.97)(1.08)(1.12) = 1.387 (38.7% total return) Manager Y: (1.15)(0.98)(1.09)(1.06)(1.04) = 1.387 (38.7% total return)

Step 2: Calculate benchmark returns Benchmark: (1.08)(1.03)(0.95)(1.06)(1.10) = 1.253 (25.3% total return)

Step 3: Analyze consistency vs. performance Both managers:

• Have identical total returns

• Beat benchmark by same amount

• Have same Sharpe ratio (given)

• Beat benchmark in 4/5 years

Step 4: Consider risk preferences Manager X has lower volatility (more consistent), but both have same risk-adjusted returns (Sharpe ratio).

Answer: The preference for Manager X is rational if the investor has strong preferences for lower volatility, even with identical risk-adjusted returns.

Investment Insight: Consistency can be valuable beyond risk-adjusted metrics for investors with strong loss aversion.

Section 5: Game Theory and Strategic Thinking

Puzzle 9: The IPO Bookbuilding Game

Scenario: You're an institutional investor in an IPO bookbuilding process. You can bid "High" ($20/share) or "Low" ($18/share). If you bid High, you get 100% allocation if the IPO is underpriced, but only 20% if fairly priced. If you bid Low, you get 50% allocation regardless of pricing. The IPO is underpriced 30% of the time, returning 25% on day one. When fairly priced, day-one return is 0%. What's your optimal bidding strategy?

Solution Process:

Step 1: Calculate expected returns for each strategy

High Bid Strategy:

• If underpriced (30%): Get 100% allocation, earn 25% return

• If fairly priced (70%): Get 20% allocation, earn 0% return

• Expected return = 0.30 × 1.00 × 25% + 0.70 × 0.20 × 0% = 7.5%

Low Bid Strategy:

• If underpriced (30%): Get 50% allocation, earn 25% return

• If fairly priced (70%): Get 50% allocation, earn 0% return

• Expected return = 0.30 × 0.50 × 25% + 0.70 × 0.50 × 0% = 3.75%

Step 2: Compare strategies High bid: 7.5% expected return Low bid: 3.75% expected return

Answer: Bid High the winner's curse is outweighed by the higher allocation when the IPO is genuinely underpriced.

Market Structure Insight: In IPO allocation games, aggressive bidding can be optimal despite adverse selection effects.

Puzzle 10: The Trading Floor Information Cascade

Scenario: Three traders make sequential decisions about whether to buy or sell a stock. Each receives a private signal (Buy or Sell) that's correct 60% of the time. They observe previous traders' actions. Trader 1 buys, Trader 2 buys. What should Trader 3 do if their private signal says "Sell"?

Solution Process:

Step 1: Analyze Trader 1's decision Trader 1 had no prior information, so buying suggests their signal was "Buy."

Step 2: Analyze Trader 2's decision Trader 2 observed Trader 1 buy and also decided to buy. This could mean:

• Their signal was "Buy" (consistent with Trader 1)

• Their signal was "Sell" but they followed Trader 1's lead

Step 3: Calculate probabilities from Trader 3's perspective Given both previous traders bought, the probability the stock should be bought is higher than the base case, even though Trader 3's signal says "Sell."

Step 4: Apply Bayesian updating Let's assume the true state is equally likely to be "Buy-worthy" or "Sell-worthy" initially.

After observing two buys, the posterior probability of "Buy-worthy" is high enough that even a "Sell" signal shouldn't override the information from previous actions.

Answer: Trader 3 should buy, following the information cascade despite their private signal.

Behavioral Finance Insight: Information cascades can lead to rational herding behavior, even when it contradicts private information.

Section 6: Interview Strategy and Tips

Approaching Probability and Logic Puzzles:

1. Structure Your Thinking

• Clearly define all events and probabilities

• Draw diagrams or trees when helpful

• State assumptions explicitly

2. Use Systematic Methods

• Bayes' Theorem for updating probabilities

• Expected value calculations for decisions

• Game theory for strategic situations

3. Check Your Intuition

• Does the answer make business sense?

• Are extreme cases handled correctly?

• What happens if you change key assumptions?

4. Communicate Clearly

• Walk through your logic step-by-step

• Explain why each step follows from the previous

• Connect mathematical results to business decisions

Common Mistakes to Avoid:

Base Rate Neglect: Always consider prior probabilities in Bayesian problems Independence Assumptions: Don't assume independence unless stated Survivorship Bias: Account for unseen failures or selection effects Anchoring: Don't let initial numbers overly influence your thinking Overconfidence: Consider uncertainty in your probability estimates

Advanced Techniques:

Symmetry Arguments: Look for ways to simplify using symmetry Proof by Contradiction: Assume the opposite and show it leads to inconsistencyRecursive Thinking: Break complex problems into similar subproblems Limiting Cases: Check what happens at extremes Alternative Framings: Sometimes rephrasing the problem reveals the solution

Conclusion

These probability and logic puzzles test the analytical skills that investment banks value most:

• Quantitative Reasoning: Ability to work with probabilities and expected values

• Structured Problem-Solving: Breaking complex scenarios into manageable pieces

• Risk Assessment: Understanding uncertainty and its financial implications

• Strategic Thinking: Considering incentives and game-theoretic aspects

• Communication: Explaining complex reasoning clearly under pressure

Success requires not just mathematical competence, but also:

• Business intuition to connect math to real-world decisions

• Logical rigor to avoid common reasoning fallacies

• Strategic thinking to understand incentives and competition

• Clear communication to explain complex ideas simply

The key to mastering these puzzles is practice, pattern recognition, and developing a systematic approach to probability problems. Remember that in actual interviews, demonstrating clear thinking and structured problem-solving is often more important than getting the exact numerical answer.