Brainteasers from Interviews - Part 4

150 Most Frequently Asked Questions on Quant Interviews

by Dan Stefanica, Rados Radoicic, and Tai-Ho Wang, second edition, FE Press 2019, 281 pages

Probability and Stochastic Calculus Quant Interview Questions

by Ivan Matic, Rados Radoicic, and Dan Stefanica, FE Press 2021, 334 pages

Challenging Brainteasers for Interviews

by Rados Radoicic, Ivan Matic, and Dan Stefanica, FE Press 2023, 345 pages

Problem 1. Let \(N\) be a random variable whose values are positive integers. Prove that \[ \mathbb E\left[N\right] ~=~\sum_{i=0}^{\infty}\mathbb P\left(N >i\right). \]

Problem 2. Each box of cereal contains a coupon. If there are \(p\) kinds of coupons, how many boxes of cereal have to be bought on average to obtain at least one coupon of each kind?

Problem 3. You roll a fair \(n\)-sided die repeatedly and sum the outcomes. What is the expected number of rolls until the sum is a multiple of \(n\) for the first time?

Problem 4. Evil Commander decided to take the cell-phones from all of his one hundred soldiers. He then correctly wrote the names of soldiers on the phones, but intentionally placed phones randomly in boxes labeled by \(1\), \(2\), \(\dots\), \(100\). One by one the soldiers are taken to the room with the boxes. Once in the room, a soldier is allowed to perform the following \(3\)-step procedure at most \(50\) times:
  • Step 1 Choose one of the boxes;
  • Step 2. Open the box;
  • Step 3. If the box contains the soldier's own cell-phone, the soldier uses the fingerprint technology to unlock it. Then he can send a message to the President voicing the discontent with Evil Commander.

After repeating the procedure at most \(50\) times, the soldier must close all the boxes and leave the room without taking any phones regardless whether the soldier succeeded in finding his/her own device.

If the President receives \(100\) messages (one from each soldier), then the Evil Commander will be required to return the phones to the soldiers. However, if at least one of the soldiers fails to find the phone in \(50\) attempts or fewer, then the President will believe Evil Commander who will deny any mischief and none of the soldiers will get their phone back.

The soldiers are allowed to discuss and decide on a strategy. Find a strategy with success rate greater than \(25\%\).

Problem 5. Seven dwarfs are imprisoned by the evil queen who decides to play the following game: The queen puts a red hat or a green hat on the head of each of the dwarfs. The hats are chosen randomly and every configuration is equally likely. The dwarfs can see all the hats except for their own. At a signal, each dwarf can stay silent, or guess the color of his hat. The queen will free all seven dwarfs if at least one dwarf guesses his hat correctly and no one guesses the hat incorrectly. If all the dwarfs are silent, or some dwarfs say an incorrect color, the dwarfs remain captured. Find a strategy for the dwarfs to go free with probability greater than \(85\%\).