Quant Interview Questions & Solutions

"Two dice. What is the probability the sum is 7?"

There are 36 equally likely outcomes (6 × 6). The pairs that sum to 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). That is 6 outcomes. So the probability is 6/36 = 1/6 ≈ 16.7%.

from itertools import product

# Q: Two dice, probability of sum = 7?
outcomes = list(product(range(1, 7), range(1, 7)))
sum_7 = [o for o in outcomes if sum(o) == 7]
print(f"Pairs summing to 7: {sum_7}")
print(f"Probability: {len(sum_7)}/{len(outcomes)} = {len(sum_7)/len(outcomes):.4f}")
print(f"= 1/6
")

"Pick 2 cards from a deck. Probability both are aces?"

First card is an ace: 4/52. Given the first was an ace, second is an ace: 3/51. Multiply: (4/52) × (3/51) = 12/2652 = 1/221 ≈ 0.45%.

# Q: Pick 2 cards, probability both aces?
# First ace: 4/52, second ace: 3/51
p = (4/52) * (3/51)
print(f"P(both aces) = (4/52) x (3/51) = {p:.6f}")
print(f"= 1/{1/p:.0f} = 1/221
")

# Verify by simulation
import numpy as np
n_sims = 1000000
deck = list(range(52))  # 0-3 are aces
count = 0
for _ in range(n_sims):
    draw = np.random.choice(deck, 2, replace=False)
    if draw[0] < 4 and draw[1] < 4:
        count += 1
print(f"Simulated: {count/n_sims:.6f} (should be ~0.004525)")

"Variance of the sum of independent random variables?"

Var(X + Y) = Var(X) + Var(Y) when X and Y are independent. The covariance term vanishes. If they are not independent, add 2·Cov(X,Y).

"Expected value of the maximum of two Uniform[0,1] random variables?"

E[max(X,Y)] = 2/3. The CDF of max(X,Y) is P(max ≤ z) = z² (both must be ≤ z). The PDF is 2z. So E[max] = ∫z·2z dz from 0 to 1 = 2/3.

"Central Limit Theorem in one sentence?"

The sum (or average) of a large number of independent random variables, regardless of their individual distributions, converges to a normal distribution.

"12 coins, one is a different weight. Find it in 3 weighings."

This is a classic 3.67-star difficulty problem. The key insight is information-theoretic: 3 weighings on a balance scale give 3³ = 27 possible outcomes (left heavy, balanced, right heavy for each weighing). You need to distinguish 24 possibilities (12 coins × heavier or lighter). Since 27 ≥ 24, it is theoretically possible. The challenge is constructing the actual scheme.

# 12 Coins Puzzle Strategy
# You have 12 coins. One is a different weight (heavier or lighter).
# Find it in 3 weighings.

def solve_12_coins():
    """
    Strategy for the 12 coins problem.
    
    Weighing 1: Compare coins [1,2,3,4] vs [5,6,7,8]
      Case A: They balance -> odd coin is in [9,10,11,12]
      Case B: Left heavy -> odd coin makes left heavy or right light
      Case C: Left light -> odd coin makes left light or right heavy
    
    Each case uses 2 more weighings to identify the coin
    AND whether it is heavier or lighter.
    
    Key insight: 3 weighings give 3^3 = 27 outcomes.
    We need to distinguish 24 possibilities (12 coins x 2 states).
    27 >= 24, so it is information-theoretically possible.
    """
    print("12 Coins Solution:")
    print("  Weighing 1: [1,2,3,4] vs [5,6,7,8]")
    print("  If balanced:")
    print("    Weighing 2: [9,10,11] vs [1,2,3]")
    print("    If balanced: coin 12 is odd, weigh vs any to find H/L")
    print("    If 9,10,11 heavy: weigh 9 vs 10")
    print("      Balanced -> 11 is heavy; else heavier one is odd")
    print("  If [1,2,3,4] heavy:")
    print("    Weighing 2: [1,2,5] vs [3,6,9]")
    print("    This narrows suspects based on which side tips")
    print("    Weighing 3: identifies exact coin")
    print()
    print("  Information theory: 3 weighings -> 27 outcomes")
    print("  Need to distinguish: 12 coins x 2 (H or L) = 24 cases")
    print("  27 >= 24, so 3 weighings SUFFICE")

solve_12_coins()

"What is put-call parity?"

This is a no-arbitrage relationship. If it is violated, you can construct a risk-free profit. It is one of the most fundamental results in options pricing and connects calls, puts, and forwards.

"What is the duration of a zero-coupon bond?"

Exactly equal to its maturity. A 5-year zero-coupon bond has a duration of 5 years. This is the maximum possible duration for any bond of that maturity, because coupon-paying bonds have durations shorter than their maturity (the coupons pull the weighted average cash flow forward).

"What happens to a call price when volatility increases?"

It increases. Always. Both calls and puts become more valuable when volatility rises because the expected payoff increases while the downside is capped at zero. This is captured by vega, which is always positive for long options.