DRW New Grad Quantitative Researcher Interview Experience — Auto-Rejected Before the OA Deadline Even Hit

I cold-applied online to DRW's US New Grad Quantitative Researcher role. Less than a week after applying, I got an online assessment invite. The email said the OA link would expire in seven days, and there was no other deadline notice anywhere. Then, five days later, I got an email saying I hadn't finished the OA in time, and I was flat-out rejected. But the OA link was still valid, so I went ahead and finished it before it expired anyway.

Since the company doesn't keep its word, I'm posting the original questions I remember here for reference. DRW sends the OA out in batches, and each batch has different questions, so you probably won't run into this exact set. But it should still be useful as a reference.

Honestly, some of these questions weren't easy — you need a pretty solid foundation in probability and stochastic processes. I wasn't well prepared and didn't answer them well.

Q1. 16 teams of distinct skill levels are randomly assigned to a fixed single-elimination bracket. The tournament has 4 rounds: 16 teams compete in 8 games, the 8 winners advance to 4 quarterfinals, the 4 remaining teams advance to 2 semifinals, and the 2 semifinal winners meet in the final. In each game, the stronger team wins with probability 90%, independently of all other games. What is the probability that the 2 strongest teams meet in the final?

Q2. A dart lands at (X, Y), where X and Y are independent normal random variables with mean 0 and variance σ². The probability of landing within 1 unit of the origin is 3/4. A throw scores 16 points if its distance from the origin is at most 1, 4 points if its distance is greater than 1 but at most 2, and 0 otherwise. What is the expected score of a single throw?

Q3. Coin A is fair, coin B lands heads with probability 80%, and coin C has heads on both sides. A coin is selected uniformly at random and tossed 3 times. Given that all 3 tosses are heads, what is the probability that coin C was selected?

Q4. A fair coin is tossed repeatedly until either HTH or HHT appears in 3 consecutive tosses. Given that HHT appears first, what is the expected number of tosses, including the final toss?

Q5. 45 stones are arranged in a circle, each holding $100. You place a token on a stone chosen uniformly at random and collect its money. You and an opponent then take alternating turns, with your opponent moving first. On each turn, the player tosses a fair coin and moves the token 1 stone clockwise for heads or counterclockwise for tails, collecting any money remaining on the destination stone. All tosses are independent, and play ends when all the money has been collected. What is the expected total amount you collect, including the initial $100?

Q6. 2 lines initially contain 6 and 7 shopping carts. In each round, a line is selected uniformly at random, and 1 cart is removed and returned to a line selected independently and uniformly at random. The cart may therefore be returned to its original line. The process ends when either line is empty at the end of a round. What is the expected number of rounds?