Optiver Interview Questions

You throw a standard die repeatedly, keeping a running total of the sums. What is the probability that when the running total exceeds 100 for the first time, the last die throw is a 2?

Six dice are rolled simultaneously. What is the probability that the sum of the numbers shown is an even number?

Provide a formal proof for the following: You have two groups of marbles—one group with 1 marble and the other with 2^n marbles. At each step, you can double the size of one group by taking marbles from the other group. Is it possible to reach a state where one group has 0 marbles and the other has 2^n + 1 marbles?

What is the expected number of card draws before encountering an Ace in a standard 52-card deck, assuming cards are drawn one at a time without replacement?

What is the expected number of moves required to exit the center of a 10x10 grid, assuming you can move one step in a random direction (up, down, left, right) at each move?

What is the probability that, given a random number from 1 through 1,000,000, the digit 7 appears at least once in the selected number?

You throw two dice and then may choose to reroll just one of them, and only once. What is the expected value?

Identify any arbitrage opportunities, if any, in the given matrix of currency conversion rates.

Can you derive the formula for ordinary linear regression?

Someone has $100 to gamble. First, he will bet $1, and if he wins, he will leave. Otherwise, he will double his bet on each subsequent round until he wins. What is his profit from this game?