← Sig Interview Insights

Sig·Software Engineer·Technical Phone Screen·Junior

Junior
May 2026

Summary

SIG quant researcher interview with a probability/expectation problem that looks clean on the surface but has a few moving parts once you start conditioning on states.

Questions Asked (1)

Q1

A spinner has three regions with landing probabilities 1/6, 1/3, and 1/2. What is the expected number of spins until the spinner has landed in at least two different regions?

Algorithms & Data Structures
Author's notes

I set it up as a Markov chain pretty quickly, which felt right.

Create a free account to read the full note

AI HintsAI Generated

Suggested Approach

Use the linearity of expectation by conditioning on the first spin's outcome. Define the expected additional spins needed from each possible first region, then solve the resulting system of equations or use the formula for the expected number of trials to get a different outcome.

Pro tip: After computing the answer, verify it by checking that the expected value lies between the minimum and maximum possible values (2 and 1 + 1/min(p_i) respectively). This sanity check catches arithmetic errors and shows rigor.

1. Define the random variable

Let X be the number of spins until at least two different regions have been landed on. Note that X is at least 2.

2. Condition on the first spin

After the first spin, we have seen one region. The expected additional spins depends on which region was seen first, so define E_i as the expected additional spins given that the first spin landed in region i.

3. Set up equations for E_i

For each region i, if the next spin is in a different region, we stop; if it is the same region, we need 1 + E_i more spins. Thus E_i = 1 + p_i * E_i, where p_i is the probability of landing in region i.

4. Solve for E_i and compute overall expectation

Solve E_i = 1/(1 - p_i). Then the overall expected number of spins is 1 + sum_i p_i * E_i, since the first spin is always needed.

5. Simplify and verify

Plug in the given probabilities: p1=1/6, p2=1/3, p3=1/2. Compute the sum and simplify. Check that the result is reasonable (between 2 and 3).

Key Points to Mention

  • Linearity of expectation and conditioning on the first spin.
  • Geometric distribution: the number of trials to get a different outcome from a fixed one follows a geometric distribution with success probability 1 - p_i.
  • The expected additional spins given first region i is 1/(1 - p_i).
  • The overall expected value is 1 + sum_i p_i * (1/(1 - p_i)).
  • Calculation: 1 + (1/6)*(6/5) + (1/3)*(3/2) + (1/2)*(2/1) = 1 + 1/5 + 1/2 + 1 = 2.7.
  • Sanity check: the answer should be between 2 and 3, and 2.7 is plausible.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.