Jane Street interview questions

At the stroke of midnight, the hour and minute hands of an ordinary analog clock lie exactly on top of one another. At what time do they next line up exactly? And over a full $24$-hour day, how many such coincidences occur in total?

A boardwalk arcade stall runs a game built around a single fair six-sided die. Two wagers are on offer. Wager 1: put down $4 on the roll landing even; a winning ticket pays $6 back in total. Wager 2: put down $6 on the roll landing odd; a win returns $9 in total. You can buy as many tickets of either type as you like. If your goal is to maximize expected profit, what is the right amount to wager?

A mobile game assigns each day's challenge by rolling a fair six-sided die: whichever face comes up, that challenge type is served. A trophy unlocks the first day you have been served all six challenge types at least once. What is the expected number of days until the trophy unlocks?

A dice bar runs a side wager: three fair six-sided dice are thrown at once and the three face values are multiplied together. What is the probability that the resulting product is even?