Two Sigma interview questions

Three assets $A$, $B$, $C$ have correlations $\rho_{AB}=0.9$ and $\rho_{BC}=0.8$. Can $\rho_{AC}=0.1$?

Sandy has $5$ pairs of socks in a drawer, each pair a distinct color. On Monday, she randomly picks two socks from the $10$ total; on Tuesday, she picks two from the remaining $8$; on Wednesday, she picks two from the remaining $6$. What is the probability that Wednesday is the first day she selects a matching pair?

Two players share a fair coin and flip it repeatedly, recording the sequence of heads ($H$) and tails ($T$) that appears. The first player wins if $HTH$ occurs before $HHT$; otherwise, the second player wins. What is the probability that the first player wins?

Ten chords on a circle have endpoints positioned uniformly and independently along the circumference. Calculate the expected number of crossing points.

14 slips numbered $1-14$ are placed in a random order. A position $i$ is called a local maximum if the slip there is strictly greater than each of its immediate neighbors. Find the expected number of local maxima. For example, with 6 numbers the arrangement 513246 has local maxima at positions 1, 3, and 6, giving 3 local maxima.

Michael rides a remote-control skateboard around campus. The front of the Hopkins sign is the origin $(0,0)$; rightward is positive $x$ and into campus (upward) is positive $y$. Every second he picks an angle uniformly from $[0, 2\pi)$ and moves 1 foot in that direction from his current position. After $16$ seconds, what is the expected squared distance from the Hopkins sign?

There are $N$ employees, each driving a separate car to QuantEssential. The cars are initially well-spaced and travel at distinct speeds assigned uniformly at random. Whenever a faster car catches up to a slower one, it adopts the slower car's speed. After a long time, the cars form $K$ clusters, each moving at a distinct speed. Find the expected value of $K$ when $N = 10$.

Marbles are drawn without replacement from a bag containing 50 red and 50 blue marbles until the bag is empty, and the sequence of colors is recorded. A run is a maximal block of consecutive marbles of the same color; for instance, $RBBRRRBRR$ has 5 runs. Find the expected number of runs in the full sequence.

In a single-elimination tournament with $2^n$ strictly ranked teams (higher always beats lower), the bracket is drawn uniformly at random. Find the probability that the top-ranked and second-ranked teams play each other in the final.

Verification. For a tournament with n = 2 (i.e., 4 teams total), what is the probability that the top two teams meet in the final?

There are $n$ identical urns, each containing white and black balls. The $i$th urn ($1 \le i \le n$) holds $1$ white ball and $2^i - 1$ black balls. An urn is chosen uniformly at random, and a ball is drawn from it; the ball is white. After replacing that ball, a second ball is drawn from the same urn. Let $p(n)$ denote the probability that the second ball is also white. Compute $\displaystyle \lim_{n \to \infty} p(n)$.

Consider a linear regression on a dataset yielding coefficients $\hat{\beta}_{\text{OLS}}$. With data matrix $X$ and IID normal errors of variance $\sigma^2$, we have $\operatorname{Var}(\hat{\beta}_{\text{OLS}}) = \sigma^2 (X^T X)^{-1}$. If the regression is rerun on a dataset where every point's values are doubled, producing coefficients $\hat{\beta}_{\text{OLS}}'$, find the constant $c$ such that $\operatorname{Var}(\hat{\beta}_{\text{OLS}}') = c \operatorname{Var}(\hat{\beta}_{\text{OLS}})$. If no such constant exists, output $-1$.

Let $x_n = [1, 2, \dots, n]^T \in \mathbb{R}^n$ and define $A_n = x_n x_n^T$. For any fixed $n$, $A_n$ has exactly one non-zero eigenvalue, denoted $\lambda_n$. Determine the constant $k$ such that $\lambda_n / n^k$ converges to a finite, non-zero limit.