EXPECTED VALUE INTERVIEW QUESTIONS

Expected value is the first thing you learn and the last thing you stop using. The interesting questions are the ones where the expectation is correct and still not the answer.

These are original representative questions, not real or leaked firm questions.

Fair value first

1. You are paid the face value of one roll of a fair die. What is the game worth?
   Show worked solution

   (1+2+3+4+5+6)/6 = 3.5. Say the number, then say that you would pay less than 3.5 to play and sell above it. Interviewers are listening for whether "fair value" and "price I would trade at" are separate ideas in your head.

2. A ticket costs $10 and pays $100 with probability 8%. Do you buy?
   Show worked solution

   Expected payoff is 0.08 × 100 = $8, so at $10 you are paying two dollars for the privilege. No. The useful follow-up you should volunteer: at what price would you buy, and how much size would you take there?

Repeated chances, and knowing when to stop

For these questions, start with the last decision. Its value tells you when to stop on the roll before it.

3. Roll a die. You may keep the value, or reroll once and keep whatever comes up. What is the game worth?
   Show worked solution

   Reroll whenever the first roll is below the value of a fresh roll, 3.5. So you keep 4, 5, 6 and reroll 1, 2, 3: (1/2)(5) + (1/2)(3.5) = 4.25.

4. Same game, but you may roll up to three times.
   Show worked solution

   Work backwards. With two rolls left the game is worth 4.25, so on the first roll you keep only 5 or 6: (1/3)(5.5) + (2/3)(4.25) = 14/3 ≈ 4.67. The method is the answer here, not the number.

5. You flip a fair coin repeatedly and win $1 for every head before the first tail. What is that worth?
   Show worked solution

   One dollar. Either sum Σ k(1/2)^(k+1), or note that the expected number of heads before a tail is (1−p)/p = 1 for p = 1/2.

Let linearity do the work

Expectations add whether or not the events are independent. That single fact turns several intimidating questions into one line.

6. You draw five cards from a standard deck. How many aces do you expect?
   Show worked solution

   5 × 4/52 = 5/13 ≈ 0.385. The draws are dependent and it does not matter.

7. Twelve people each throw one hat into a pile and take one back at random. How many get their own hat on average?
   Show worked solution

   One. Each person has a 1/12 chance, and 12 × 1/12 = 1. The answer is one for any group size, which is the part that makes it a good question.

When expected value is not the answer

These two bets have the same expected payout. Their risks are very different.

8. Bet A wins $1 with certainty. Bet B wins $1,000,000 with probability 0.000001 and nothing otherwise. Same expected value. Are you indifferent?
   Show worked solution

   A risk-neutral player could be indifferent. Someone who prefers a guaranteed payout could choose A. Explain your objective, bankroll and tolerance for risk; equal expected values alone do not settle the choice.

9. You price a contract at 50 and someone immediately buys 500 at your offer without negotiating. Is your expected value still 50?
   Show worked solution

   It may change. If the buyer is informed, eagerness to buy can suggest the payoff is worth more than your estimate of 50, making your sale less attractive. If the buyer is uninformed, the trade may not change expected payoff at all. Explain the adverse-selection assumption before re-quoting.

Where to go next

Expected value is half of a market-making answer; the other half is turning it into a two-sided quote. That is covered in the market-making guide. For the probability machinery underneath these, see the probability question bank.

Explain why your decision follows

For the die stopping problems, assume fair independent rolls, no reroll cost and a goal of maximizing expected payout. With a cost or a different objective, the optimal stopping threshold changes. Positive expected profit alone does not specify a sensible position size. The effect of a customer trade depends on whether their action conveys information.