Optiver Interview Guide

At a quiet border crossing, the probability that at least one vehicle shows up during any $20$-minute window is $0.36$. Vehicles arrive with constant, memoryless randomness, so arrivals in disjoint time intervals are independent. What is the probability that at least one vehicle shows up during a $10$-minute window?

Two fair six-sided dice are thrown together, and a dealer pays you, in dollars, the larger of the two face values (if they tie, you're paid the common value). What is the expected payout of this game?

Two fair six-sided dice are rolled. Lena is paid, in dollars, the smaller of the two face values; Marco is paid the absolute difference between the two faces. Whose expected payout is larger, and by exactly how much?

A colleague flips a fair coin $10$ times, writes down the sequence, and truthfully reports one fact: at no point did two heads come up back-to-back. Given that report, what is the probability that the very first flip was heads?

A clearing house settles contracts only in lots of $77$. A client needs the smallest settlement amount strictly greater than $70{,}000$; a second client needs the smallest strictly greater than $700{,}000$. Find both multiples of $77$ — mentally, and fast.

An interviewer offers you game A: a one-shot bet with an expected profit of $+\$5$ and a standard deviation of $\$20$. You take it, and you tell him it is right at the edge of what you would accept -- any less edge and you would pass.

He then offers game B, also one-shot and independent, take-it-or-leave-it at a size he sets, with a standard deviation of $\$50$.

(a) If you price risk consistently, what expected profit does B need before you take it? (b) Your friend says B needs only $+\$12.50$, "because that is the same reward per unit of risk." Who is right? (c) What is actually being tested here?

A bookmaker posts decimal odds on a match with exactly three outcomes -- a stake of $\$1$ at decimal odds $d$ returns $\$d$ in total if the outcome hits, and nothing otherwise.

$$\text{Home } 2.10, \qquad \text{Draw } 4.00, \qquad \text{Away } 4.20.$$

(a) Is there a risk-free profit here? Show how you know in one line. (b) With $\$1{,}000$ to deploy, how do you split it, and what do you make? (c) A second book posts odds whose implied probabilities sum to $1.05$. What does that number mean, and why is the situation in (a) so rare in practice?

A standard $52$-card deck is well shuffled and dealt face up one card at a time.

(a) What is the expected number of cards dealt before the first ace appears? (b) What is the expected position of the first ace, and of the last ace? (c) Answer without summing a series.

Three cards carry three distinct numbers you know nothing about -- no range, no distribution. They are shuffled and placed face down, and you turn them over one at a time. After each card you must either stop on it or discard it permanently; if you discard the first two, you are stuck with the third.

You win only if the card you stop on is the largest of the three. You can compare the values you have seen, but the numbers themselves tell you nothing about what is still face down.

What is the best strategy, and what is your probability of winning? Beat the two obvious answers of $\tfrac{1}{3}$.

Two players each flip a fair coin $4$ times, independently of one another. What is the probability that they end up with the same number of heads?

Then generalise: with $n$ flips each, roughly how does that probability behave for large $n$?