Optiver Interview Questions

A family has children one by one until the number of daughters exceeds the number of sons by exactly one. What is the expected total number of children they will have?

You roll three dice, remove any pair duplicates, and get paid the final sum. How much would you pay for a call option on this payout with a strike price of $14?

What is the probability of getting 10 heads in a row when flipping a fair coin?

Given a bracket of teams and their relative strengths or ratings, determine how to allocate bets on the teams to maximize expected profit and loss (PnL).

Calculate the value of 1.02 multiplied by (19.25 minus 0.36).

In a perfectly efficient market, there are two companies: A with a success probability of 2/3 and B with a success probability of 1/2. Both companies are initially worth $50 each. If both companies either succeed or fail, nothing special happens. However, if one company succeeds and the other does not, then the company that does not succeed goes bankrupt. How would the price of each company change if both succeed?

What is 3 plus 5 equal to?

How would you price an asset, given a list of bids and asks?

How many cards are needed to build a 100-storey house of cards, assuming each storey is constructed in the standard way?

Estimate how many digits are in 100 factorial (100!).

Given a stream of stock price updates for both buy and sell orders, develop an efficient method to send and process the data to reduce overhead. Discuss considerations for using TCP versus UDP for transmitting this data.

Out of 100 fair coin flips, what is the probability that you get at least 10 more heads than tails?

Implement a linear regression algorithm from scratch, without using any machine learning libraries.

Design a double-ended queue (deque) data structure that supports efficient insertion and deletion from both ends. Explain the operations and their time complexities.

Describe how you would implement a queue data structure.

How would you implement a queue? As a follow-up, actually build a queue using an array.

What is the difference between UDP and TCP?

Given a string representation of a binary tree, write a function to parse the string and construct the tree. Then, identify and return a list of errors found in the tree, such as nodes with more than two children, more than one root, or duplicate parent-child pairs.

What is memory fragmentation, and how can it impact performance?

1. What is 73 × 87? Provide your answer without using a calculator and explain your reasoning. 2. What is the probability of drawing exactly two aces from a shuffled deck of cards?

In a sequence of fair coin tosses with each outcome being either Heads (H) or Tails (T), what is the probability that the sequence HHH appears before HTH?

How many nodes are there in a recombining binomial tree with n steps? How many paths lead to a particular node?

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?

Given a list of exchange rates between different currency pairs and an amount in one currency, convert the amount to another specified currency.

Create functions that simulate communication with exchanges and ensure that short-selling is avoided.

Given a graph and two vertices, implement Dijkstra's algorithm to find the shortest distance between the pair. Additionally, return an error if there are multiple shortest paths with the same distance between the vertices.

Given a mapping of 7 letters to their respective Morse code representations and a string of dots and dashes (with no spaces), find all valid interpretations of the Morse code string as sequences of those letters.