Jane Street Interview Questions

If you roll a fair coin 10 times, what is the expected value of the product of the number of heads and the number of tails?

What is the expected value of the product when rolling a six-sided die and an eight-sided die?

Estimate the mass of the Earth.

If it is 12 o'clock right now, when was the last time the hour and minute hands of the clock were aligned?

At 12:00, the hands of a clock are perfectly aligned. Assuming continuous motion, at what time will they be aligned again?

What quantitative characteristics would you use to qualitatively assess the difficulty of a maze?

What is 19% of 19?

On a bus with three stops, at each stop, three-quarters of the passengers get off and seven new passengers board. What is the minimum integer number of passengers that must have been on the bus at the start so that the number of passengers after each stop is always an integer?

Flip a fair coin repeatedly until either the pattern HHT or HTT appears. What is the probability that HHT appears before HTT? Please elaborate your reasoning.

Today is 15/09/2011. What is the nearest future date such that all eight digits in the date (DDMMYYYY) are unique?

What is the last digit in the value of 3 raised to the power of 33 (i.e., 3^33)?

What is 29 multiplied by 33? Solve it without using pen and paper.

What is 1,000,000 minus 112, without using pen and paper?

If I have a 4x4x4 cube and I paint the outside of the cube, then cut it into 64 1x1x1 cubes: 1) How many of the smaller cubes have at least one side painted? 2) If I pick one of the 64 cubes at random and roll it, what is the probability that the top side is painted?

If I roll two dice, what is the probability that the second die shows a higher number than the first?

You dip a cube into paint and let it dry. Then you cut it into 27 smaller sub-cubes and put them in a bag. If you randomly pick one sub-cube and roll it on the table, what is the probability that no paint is visible?

In an infinite sequence of coin flips, what is the probability that HHT appears before THH?

In a World Series (baseball) between two teams, A and B, each with a 50% chance of winning any game, and the first team to win 4 games wins the series, what is the probability that the series reaches game 7 (i.e., each team wins 3 games before the final game)?

There are 10 lightbulbs in a row, each of which can be either on or off. No two adjacent lightbulbs may be on at the same time. How many possible combinations are there?

If you roll a fair coin 12 times, what is the expected product of the number of heads and the number of tails?

You have 100 green marbles and 100 red marbles and can divide them however you like between two bags, with at least one marble in each bag. A player, who does not know the distribution, randomly chooses one of the bags and draws a single marble. If the marble is green, the player wins $100; if red, they win nothing. What is the fair price for this game?

There are 4 coins. You earn a dollar for every head. All four coins are flipped. What is the expected payout? Additionally, if you have the option to either keep the result of the first flips or reflip all four coins, what is the optimal expected payout?

Three coins are tossed. What is the probability of getting at least two heads?

X is uniformly distributed on [0, 5]. What is E(|5 - X|) if X is continuous vs. discrete? Which value is larger, and why?

How would you simulate the outcome of a six-sided die using only a fair coin?

What is the expected number of coin flips required to simulate a fair 6-sided die?

You toss 4 coins. For each head that appears, the player receives $1,000. The player is then given a second chance to play the game again. What is the optimal strategy for the game, and what is the expected monetary value of the game?

You have a box filled with cash, where the cash value is uniformly distributed between 1 and 1000. You participate in an auction: you win the box if you bid at least the value of the cash inside; you win nothing and lose nothing otherwise. If you win, you can resell the box for 150% of its value. How much should you bid to maximize the expected value of your profit (resale value minus bid)?

Calculate 15% of 155. You have 10 seconds.

What is 76 multiplied by 84?