Source: Problem prompts adapted from Ace the Data Science Interview — Chapter 5 Hard set (Huo & Singh). Solutions and explanations here are original for this course.
Seven advanced probability interview questions, continuing from the easy and medium sets. Same tile format: click a problem, enter a numeric answer. Green tiles are ones you've solved, and progress is saved in this browser. If you check an answer wrong twice, the official answer and explanation are revealed automatically.
Hints: these mix combinatorics with a distributional model (binomial, normal) or a branching/absorption argument — pin down the right model before grinding through algebra.
0/7 solved
5.29D.E. Shaw
You have 150 friends. Each friend's last-four phone digits are equally likely to be any of the 104 combinations, independently. What is the probability that exactly 3 of your 150 friends have last-four-digits that are some permutation of 0, 1, 4, and 9?
Enter a number (at most 4 decimal places).
5.30Spotify
A fair six-sided die is rolled n=5 times. What is the probability that the largest value rolled is exactly r=4?
Enter a number (at most 2 decimal places).
5.31Goldman Sachs
A jar starts with a single amoeba. Every minute, each living amoeba independently has a 1-in-4 chance of each of: dying, doing nothing, splitting into 2, or splitting into 3. What is the probability the jar eventually has no living amoeba?
Enter a number (at most 2 decimal places).
5.32Lyft
A fair coin is tossed n=10 times, landing on k=4 heads. What is the probability that the first toss was heads?
Enter a number (at most 2 decimal places).
5.33Quora
You take N=5 i.i.d. draws from a Normal(μ=0,σ=1) distribution. What is the probability that exactly k=2 of the draws are larger than Y=1?
Enter a number (at most 2 decimal places).
5.34Akuna Capital
You pick three points independently and uniformly at random on a unit circle and form a triangle from them. What is the probability the triangle contains the circle's center?
Enter a number (at most 2 decimal places).
5.35Citadel
A bag has r=3 red balls and w=7 white balls. You draw balls one at a time, without replacement, until the bag holds only one color. What is the probability that white balls run out first?