There was an initial probability-based word problem, to be solved purely mathematically without code. After this, there was a very open-ended product case study, with a "working backwards document" as the final deliverable.

Do not apply. Heed the warning of all the other reviews. First, I was given a taken home which I completed. This was the probability problem as described by others (Not the cab hailing which was previous I guess). After an interview, he asked me to explain my solution to the first take home. He himself could not understand my solution and I am not sure why he is reviewing it. After the interview, he gave me yet another take home. I am not going to do it because it is a current problem they are having. They are simply looking for free consultant advice and I doubt even if you did a stellar job that you'd get hired. They want you to go deep into their 'real' data and find customer problems, find a solution and write a detailed paper. My thinking is, if you want free consultant advice my rate is $250 an hour. Don't fall for this scam (as others have already said and I didn't heed the warning and did the first take home) because they are just looking for free advice. You'd be a fool to do that many take home assessments. The second one is really involved and I would think it would take about 5-10 hours. Again, they are not looking for trivial data solutions but to go "all-in", as I was told. If you were a consultant, you'd probably charge 20k to give them such a solution along with a paper that describes your thinking (which is what they want and you'll get a follow up interview where they ask you questions). This is expertise gouging and should be against the law!