Jane Street Interview Questions

If I flip four coins and give you a dollar for every head, what is the expected value of this game?

Given 99 coins (not necessarily fair) and 1 fair coin, what is the probability that the total number of heads flipped is even?

A fair coin is flipped repeatedly. What is the expected number of flips required to obtain three consecutive heads (HHH)? What is the expected value of the game?

Two players each choose a different integer between 1 and 30. Then, a random integer x between 1 and 30 is generated. The player whose chosen number is closer to x receives x dollars. Analyze the optimal strategies for each player.

What is the minimum number of people required to guarantee that at least 5 people share the same birthday month?

(i) How many 7-digit numbers are there in the form abcdcba (symmetric about the middle digit)? (ii) What is the average value of all such numbers?

What is the sum of all odd numbers between 1 and 80?

You throw two eight-sided dice. What is the probability that the product of the two numbers shown is even?

What is the last digit of 17 raised to the power of 17?

What is the probability that a series of fair coin flips results in an even number of heads? What is the probability if the coins are not all fair, but a subset are?

What is the fair price of a game in which you shuffle 9 cards labeled 1 through 9, and then repeatedly choose whether to open the next card or stop, receiving the sum of the last decreasing sequence of cards revealed?

The chance of rain on Saturday is 50%, and the chance of rain on Sunday is 60%. What is the probability that it will rain on at least one day during the weekend?

If I flip a fair coin five times, what is the probability of getting at least two heads?

What is the angle between the hands of a clock at 9:30?

What is the probability of flipping at most two tails when flipping five fair coins?

What is the minimum number of people required to ensure that at least 7 of them share a birthday in the same month?

You roll a fair six-sided die. If you roll an even number, for every $2 you bet, you win $3. If you roll an odd number, for every $4 you bet, you win $6. What is your expected return, and how should you bet?

Given 100 coin flips, what is the probability of getting an even number of heads?

Two painters work at different rates. How long will it take them to paint a room if they work together, given their individual painting rates?

If you throw a fair die with 6 faces repeatedly, what is the expected number of throws until you get two consecutive sixes?

There are 10 people in a room. If each person shakes hands with every other person exactly once, how many total handshakes are there?

If you flip 4 fair coins, what is the expected number of heads?

You throw a fair die and are paid the number you roll. If you are not satisfied with the first result, you may choose to throw a second time and must accept the result of the second throw. What is the expected payoff?

Is Simpson's paradox possible? If so, provide an example illustrating the paradox.

What day of the week will it be exactly one year from today?

What is the probability of drawing two queens from a standard deck of 52 playing cards?

A cube painted with red paint is divided into 27 smaller cubes (3×3×3). If one small cube is randomly selected and tossed, what is the probability that at least one of the sides that are visible has paint on it?

You are given a six-sided die. What is the expected value of the difference between two dice rolls?

If I roll a pair of dice and know that one die shows a six, what is the probability that both dice show six?

I am thinking of two positive integers, a and b. I can tell you that a/b belongs to the closed interval [0.48, 0.52]. Give all possible values for b, no matter what value a takes.