Jane Street Quantitative Trading Intern Interview Experience — Three Phone Rounds, All Game Theory

I'll share my experience with the three phone-interview rounds for a quant trading internship at Jane Street's NYC office. Jane Street likes to test with games — no math brainteasers, no algorithms, none of that standard stuff. You don't need to answer fast either; taking a minute or two to think or calculate before giving an answer is completely fine.

Round 1: two questions, both original questions from Glassdoor.

Question 1 — The game runs for 100 rounds. We have two bottles and a bunch of balls. Each round you either randomly put a ball into one of the bottles (you don't know which one it went into), or randomly take a ball out of one of the bottles (again, you don't know which one). The goal is to take out as many balls as possible. What's the best strategy, and what's the expected number of balls you take out under that strategy?

Question 2 — You flip a coin 100 times. The probability that the coin lands heads follows Unif(0, 1). Each time you guess which side it'll land on, and a correct guess pays you $1. If it costs $x to play this game, would you be willing to play? What strategy would you use each time to guess which side?

Round 2: You roll a ten-sided die. If the number is 2 through 10, it gets added to your winnings. If it's a 1, your winnings reset to zero. When should you stop rolling? Now you play this game together with another player, and whoever ends up with higher winnings gets their final winnings paid out by a third party. What's the best strategy?

Round 3: They started with some resume and behavioral questions — what I like to do in my free time, what games I play, that kind of thing. Then came a question I'd seen before on the forum: you roll two dice. You can only see one die's number, and the other player can only see the other die's number. You take turns bidding on the sum of the two dice. What's the best strategy?

There was a 1-2 week gap between each round. HR responds fast — you basically hear back about the next round within two business days of finishing an interview.