Jane Street Interview Questions

You have two dice: one 10-sided and one 6-sided. You may guess any number between 2 and 16. If the sum of the two dice equals your guess, you win that number of dollars. What number should you guess to maximize your expected winnings?

Suppose you are standing at one-third the length of a bridge when you hear a train coming from behind. You have just enough time to run back and get off the bridge before the train reaches you, and you also have just enough time if you run forward. What is the relative speed between you and the train?

1) What is 12% of 47? 2) If I pick 2 cards from a shuffled deck (with no jokers), what is the probability that both are queens? 3) i) If I toss 4 fair coins and earn a dollar for every head, what is the expected value of this game? ii) If I can re-toss all 4 coins and must take the value from the second round, what is the expected value now? 4) How confident are you that you got 0, 1, 2, 3, or all questions correct, giving percentages? 5) If two teams play a best 4 out of 7 series, where each team has a 50% chance to win each round, what is the probability that the series lasts all 7 games?

What is one million minus one hundred eleven?

What is the result of multiplying 567 by 39?

What is the probability that if you roll two dice, the sum is greater than 7?

You roll a six-sided die and receive the amount you roll in cash. How much should you pay to play this game? What if you do not like your first roll and are allowed to roll exactly once more (but you must take what you get the last time)? What if you can roll up to three times, always keeping the last roll? What if you can roll infinitely many times?

A cube that is white on the inside is painted blue on the outside and then cut into thirds along each dimension, resulting in 27 smaller cubes. Blindfolded, I randomly pick one of the smaller cubes and toss it so it lands with one side down. I observe that all of the visible sides are white. What is the probability that the side facing down is blue?

You have a tricycle and plan to travel one thousand miles. You also have two spare tires with you. If you want each of the five tires to be worn equally by the end of the journey, what is the minimum number of stops you must make to achieve this?

It is currently 10:02. What is the angle between the hour and minute hands on a clock?

In a best-of-7 game between two players, you wish to place 50/50 bets before each point such that if player A wins the series you win £1000, and if player B wins the series you lose £1000. How can you structure your bets to achieve this outcome?

How many times does the digit '1' appear in all the numbers from 1 to 1,000,000 inclusive?

Sum all the odd numbers between 1 and 100.

Estimate the weight of Mount Kilimanjaro.

Consider a game where you roll a fair six-sided die. What is the expected value of a single roll? What is the expected value if you can roll twice and either keep the result of the first roll, or discard it and are forced to keep the second roll? How does this extend to three rolls or an infinite number of rolls?

What is the result when you subtract 12 from 1,000,000?

What is 2 to the power of 10?

If I spin a roulette wheel, roll a die, and pick a card from a 52-card deck, what is the probability that all three show the same number? How confident are you in your answer?

If you flip ten coins, what is the expected value of the product of the number of heads and the number of tails?

If you have only 5-cent and 11-cent stamps, what is the smallest amount that cannot be formed using any combination of these stamps?

a) What is the expected value of a die? b) Suppose you play a game where you receive a dollar amount equivalent to the number of dots that show up on a die. You roll once; if you don't like the result, you may reroll, but you must keep the second roll. What is the fair value of this game? c) Same as (b), but now you may reroll up to twice.

You have a 6-sided die and a 10-sided die. You roll both dice together, guess the sum, and if you guess correctly, you win that amount in dollars. What sum should you pick to maximize your expected winnings?

Six cups and saucers come in pairs: there are two red, two white, and two with stars. If the cups are placed randomly onto the saucers (one on each), what is the probability that no cup is placed on a saucer of the same pattern?

You have 4 fair coins. When you toss the coins, you win an amount in dollars equal to the total number of tails. Additionally, you have the option to re-toss all the coins, but this costs you 1 dollar. What is the expected value of this game?

What is the probability of getting exactly 2 heads in 4 coin tosses?

One player rolls three six-sided dice (3d6), while the other player rolls one twenty-sided die (1d20). Which player has the higher probability of obtaining the greatest score?

What is the probability of getting an even number of heads after tossing a fair coin n times?

What is the expected value of the result of a single fair six-sided die roll?

What is the probability of getting an even number of heads when tossing 4 coins? How does this change with 100 coins? What if one coin is unfair? What if all coins are unfair? What if only one coin is fair? How many fair coins are needed to ensure a 50% chance of getting an even number of heads?

Player A rolls three six-sided dice (d6) and sums the values, while Player B rolls one twenty-sided die (d20). Which player has a greater probability of getting a higher number?