Probability & Expected-Value Screen
A free, instantly-graded run through the dice, coins and betting-game questions that open quant and trading first rounds.
The scenario
It's the first technical screen for a quant role. Before anything else, the desk wants to see you can reason about randomness quickly and correctly the kind of dice, coin and betting-game questions that fill Optiver's Beat-the-Odds and Jane Street's phone screens. Commit to a number, then check it.
Where this shows up
Rapid probability and expected-value questions are the single most common quant/trading first-round screen usually timed, multiple-choice or short-numeric, with answers you reach by reasoning rather than heavy calculation.
Firms such as Optiver, Jane Street, SIG.
DeskPrep is not affiliated with, endorsed by, or sponsored by any named firm. Firm names are used for illustrative, educational purposes only and do not imply that these materials are official assessments of, or are connected with, those firms.
The drill
Answer each question. Numeric answers are graded to a small tolerance, so reasonable rounding is fine. Decimals between 0 and 1 are probabilities.
- 011 pt
You flip a fair coin until the first heads appears. What is the expected number of flips?
- 021 pt
Roll two fair six-sided dice. What is the probability the sum is exactly 7? Give a decimal between 0 and 1.
How many of the 36 equally likely outcomes sum to 7?
- 031 pt
Roll two fair dice. Given that the sum is at least 9, what is the probability that at least one die shows a 6? Give a decimal between 0 and 1.
- 042 pts
You are paid, in pounds, the face value of one roll of a fair die but you may re-roll once and must keep the second roll. What is the fair value of this game?
When should you re-roll? Compare each first-roll outcome to the value of re-rolling.
£ - 051 pt
You flip a fair coin until you get two heads in a row. What is the expected number of flips?
- 061 pt
Roll a fair die twice. What is the probability the second roll is strictly greater than the first? Give a decimal between 0 and 1.
- 071 pt
A game: roll two dice; you win £21 if the sum is exactly 7, otherwise you pay £5. Should you play?
- 081 pt
What is the expected value of that bet, in pounds? (Win £21 with probability 1/6, otherwise lose £5.)
£ - 091 pt
You remove the top 10 cards of a well-shuffled standard 52-card deck without looking at them. What is the probability the next card is red?