SP-1 / Lesson 6
This not a homework in any way, at least because the course has come to an end... The batches with questions from below server one purpose: given you another set of practise mathematical questions for the interviews. We recommend, you time yourself and try to solve them within the specified time range.
Good luck with your future endeavours and maybe see you at the SP-2 course?
Take a sit at the table and have a pen & a piece of paper ready. Then pick any of the batches and start working on them, don't forget to time yourself! Just as the problems from the first problem set, these questions have actually come up during an interview to one of the companies mentioned in the classwork.
Problem 1.
Quickly calculate $76 \cdot 84$.
Problem 2.
If the probability of seeing a shooting star over the course of an hour is 0.64, what is the probability of seeing a shooting star over the course of a half hour?
Problem 3.
Given a race track with 5 lanes, 25 bunnies, and no timer, how many races are required to find the top 3 fastest bunnies?
Problem 4.
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. What is the probability that she will roll the die an odd number of times?
Problem 5.
There are $n$ independent random variables $X_1, ..., X_n$ with each being a uniform $[0,1]$. Let $Y = \min(X_1, ..., X_n)$ and let $Z = \max(X_1, ..., X_n)$. What is $\mathbb{E}[Y]$ and what is $\mathbb{E}[Z]$? If your calculations are correct, you should get that $\mathbb{E}[Y] + \mathbb{E}[Z] = 1$. Does it make sense?
Problem 6.
We start with an urn containing $n$ white balls and $m$ black balls. The evolution of the urn occurs in discrete time steps. At every step a ball is chosen at random from the urn. The color of the ball is inspected and then the ball is discarded. What’s the probability of getting a white ball in the $k$-th step?
Problem 1.
You have 0.25 chance to forget an umbrella in a store. You found you forgot an umbrella after visiting two stores. How much is the chance you forgot it in the first one?
Problem 2.
My telephone rings 12 times each week, the calls being randomly distributed among the 7 days. What is the probability that I get at least one call each day?
Problem 3.
Choose two random numbers from $[0,1]$ and let them be the endpoints of a random interval. Repeat this $n$ times. What is the probability that there is an interval which intersects all others?
Problem 4.
You are playing a one-player game with two opaque boxes. At each turn, you can choose to either "place" or "take". "Place" places $1 from a third party into one box randomly. "Take" empties out one box randomly and that money is yours. This game consists of 100 turns where you must either place or take. Assuming optimal play, what is the expected payoff of this game? Note that you do not know how much money you have taken until the end of the game.
Problem 1.
What’s the expected number of tosses needed to get a first consecutive 2 heads? Same question for getting THHTHT.
Problem 2.
Estimate the probability of exactly 50 heads in 100 tosses of a coin. What is the probability of exactly 55 heads?
Problem 3.
There are $n$ data points $(x_i, y_i)$ and suppose that you fit in a simple regression minimising LSE. Will this LSE necessarily increase/decrease if you delete one of the data points?
Problem 4.
A point $P$ is chosen at random inside an equilateral triangle. What is the probability that the perpendiculars to the sides of the triangle from $P$ can be arranged to form a triangle? (Hint: think about the areas and the sum of lengths of these perpendiculars)