Probability and Expected Value Problems

This is a compilation of the resources I have used for interviewing with Jane Street in Fall 2022.

The first link here: Useful Advice is some good general advice to read before going into these type of interviews to know what to expect.

Mental Math ZetaMac is a great resource for getting sharper at mental math. Unfortunately, during my interview with Jane Street there was no mental math involve.

Expected Value + Recursion Problems 100 die-problem Actual Interview Question

In the Anna and Stairs Problem, we get a 3rd order recurrence relations

$E(n) = \frac{1}{2}(E(n-1) + 1) + \frac{1}{2}(E(n+1) + 1)$

and

$E(n+1)= \frac{1}{2}(E(n)+1) + \frac{1}{2}(E(n+2) + 1)$

We can get the equation

$E(n+2) - 3E(n+1) + 3E(n) - E(n-1) = 0$

We can use our initial conditions of: $E(99) = 0$ and $E(1) = 1 + E(2)$ Dice games Conditional Probability Confidence Intervals Markov Chains

Prob + Stat Pidgeonhole Principle

We can use symmetry to find way to prove certain principles.

We can derive hidden rules from the initial conditions of our problem

150 Interview Questions

Actual Interview Question

Compilation of Questions

On Learning Combinatorics

Expected Value and Markov Chains

Harvard Stats

Question Compilation:

Bidding games:

A number from 0 to 1 is put on a piece of paper. You and person X are playing a game. You must get less than the number on the paper, but above X’s guess. You know X chooses to pick a number at random. What is lowest number you can choose to maximize your probability of winning

Suppose you are given the opportunity to bid for a treasure chest, which you know with 100% confidence to be priced anywhere between $0-$ 1000. If you bid equal to or above the price, you win the treasure chest (at the cost of your bid). If you bid below the price, you do not earn the treasure chest. Now, also suppose you have a friend who is willing to buy the treasure chest from you for one and a half times the price of the treasure chest (should you obtain the chest). What should your bid be?

You have a drawer with an infinite number of two colors of socks, which exist in equal probability. What is the expected number of attempts at taking out socks individually from the drawer before a matching pair is found?

There is a raffle with 80 total tickets. Three tickets win a prize. I buy 5 tickets. What is the approximate probability that I win exactly one prize?

Tips: Use expected value Come up with an expression based on the random variable selected Seek to maximize or minimze the expression. Use probability rules Patterns regarding the state transitions and exploiting them (HARD and need better intuition)