First, just to be clear: this is an interview for a full time trader role, and I was not an intern at Jane Street (nor any other trading firm). My interview process was as follows: I had three phone screens, of approximately increasing difficulty. Each was with a trader, and every interviewer was extremely kind, friendly, and helpful. Each of the interviews included probability/expectancy questions. I had far too much trouble with one question in the second round, but the interviewer was helpful on guiding me on the right track and I managed to come up with the correct answer soon after. The second phone screen included an estimation questions, and I was asked for a confidence interval (I looked up the answer later and I was horrendously off anyway, but still moved forward). The third interview involved the woman asking me for the expected payout for a die-related strategy game with optimal strategy. I wasn't given nearly enough time to come to the optimal answer (which she acknowledged), and was asked to give a 90% confidence interval. I came up with my interval, and we discussed it at length. She eventually offered me a favorable bet, and after I said I would be happy to accept, she asked what the chances were that she actually knew the correct answer and that I was wrong!! I gave her a high chance, and she warned me about the phenomenon of adverse selection. She did, however, pass me to the final onsite interview.
The onsite interview was in the New York office. As other have mentioned, it did consist of a series of betting games that focused on making markets. I began with a hundred chips with which to offer to buy and sell. For example, you may make a market on the sum of four standard die rolls. He then opts to either buy or sell against your sell or buy price. Then, as he rolls one die, he continues to offer to buy and sell on various markets. For example, maybe the product of the two highest rolls, the product of the lowest three rolls, the maximum roll, and so on. The interviews continued like this for the entire day, and I ended with almost 200 chips. After the five interviews/seven hours, I had to leave and catch a train. The very next day, I was emailed and asked to come back for more interviews. I came back and interviewed with three or four more folks from the specific desk I would be working on. For the first time, I received some behavioral/fit questions (although I am sure I was being evaluated before), and there were no more betting games. The final person to come in gave me the offer! We spoke in detail from then on about the specifics of the offer as well as what my role would be at Jane Street.
All in all, it was an incredibly fun, challenging, and ultimately rewarding experience. All of the interviewers were nothing but helpful, friendly, and positive people - even when I got stuck. One interviewer was intentionally disputatious, but he made sure I knew that he was role-playing. However, each interviewer was definitely honing in on the details and making sure I understood exactly what I was saying. This is not a group of people on whom you can pull fast ones. They are all sharp, astute, and erudite. Good luck to anyone