Expected value questions in trading interviews, with worked answers

Expected value is the language of trading interviews. Almost every game, bet or market-making question reduces to: what is this worth on average, and what would you pay?

The definition

The expected value of a bet is the sum of each outcome multiplied by its probability. A fair price is the expected payout.

Worked questions

1. You roll a die and are paid the number shown. What is the game worth? (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5.

2. You may reroll once. What is it worth now? Reroll when the first roll is below 3.5, so on 1, 2 or 3. Half the time you keep an average of 5; half the time you get 3.5. 0.5 × 5 + 0.5 × 3.5 = 4.25.

3. Two rerolls? With two rolls left the game is worth 4.25, so keep only a 5 or 6 on the first roll. (1/3) × 5.5 + (2/3) × 4.25 = 4.67.

4. You are paid the larger of two dice. P(max = k) = (2k − 1) / 36. The expected value is 161 / 36 = 4.47.

5. You flip a coin until the first tail and get 1 for each head. The number of heads has mean 1.

6. A bet pays 3 to 1. What probability makes it fair? 3p − (1 − p) = 0, so p = 1/4.

7. A stopping game. You roll a die and add up the rolls. You can stop any time, but a 1 wipes out your total. When do you stop? Rolling again at total s is worth (5/6)s + 20/6. That beats s only when s is below 20.

Techniques

• Linearity. The expected value of a sum is the sum of expected values, even when the parts are dependent.
• Work backwards. For games with choices, value the last decision first.
• Condition on the first step. Split on the first roll or flip and write an equation.
• Symmetry. Look for outcomes that must be equally likely.

What interviewers add

• "What would you pay to play?" Slightly less than the expected value, and say why: variance and bankroll.
• "Now make me a market." Quote a bid below and an offer above your fair value.
• "I buy from you. Does your price change?" It should, if the trade tells you something.

Common traps

• Averaging the wrong thing, such as average of a ratio.
• Ignoring the option to stop or reroll.
• Forgetting that paying to play changes the profit, not the payout.

Practise timed sets in trading prep and read trading brainteasers.