How do I approach Statistics & Math interview questions?

Statistics & Math questions require understanding of core concepts and practice. PracHub provides solutions with explanations to help you master statistics & math interviews.

What difficulty level is this interview question?

This is a easy difficulty Statistics & Math question, commonly asked during Technical Screen rounds at Upstart.

What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Upstart during technical interviews.

Calculate Particle Survival Probability After Time t

Quick Overview

This interview question evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer for Calculate Particle Survival Probability After Time t states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

Calculate Particle Survival Probability After Time t

Radioactive-Decay Style Probability

Context

You have 100 identical, independent particles. Each particle's lifetime is exponentially distributed with rate parameter λ corresponding to a known half-life H. For an exponential lifetime, the survival function is:

P(T > t) = e^{-λ t}, where λ = (ln 2) / H.

Task

Derive the probability that exactly k particles (out of 100) remain undecayed after time t.
Derive the probability that at least one particle remains undecayed after time t.

Assumptions

Particles decay independently.
All particles share the same half-life H (i.e., the same exponential rate λ).
Time t ≥ 0 and k ∈ {0, 1, ..., 100}.

Constraints & Assumptions

Preserve the scope, facts, inputs, and requested outputs from the prompt above.
If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it.
Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate.

Clarifying Questions to Ask

Clarify the random variables, distributional assumptions, independence assumptions, and desired output.
Show enough derivation for the interviewer to follow the reasoning.
Explain how you would validate the result with simulation or sensitivity checks.

What a Strong Answer Covers

A correct setup with definitions, formulas, and boundary conditions.
A step-by-step derivation or estimation plan.
Interpretation of the result, including uncertainty and practical limitations.
Checks for assumptions, edge cases, and numerical stability.

Follow-up Questions

How would the result change if the assumptions were relaxed?
Can you verify the answer with a simulation?
What is the most likely source of estimation error?

Quick Overview

Assumptions

Particles decay independently.

All particles share the same half-life H (i.e., the same exponential rate λ).

Time t ≥ 0 and k ∈ {0, 1, ..., 100}.

Constraints & Assumptions

Preserve the scope, facts, inputs, and requested outputs from the prompt above.

If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it.

Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate.

Clarifying Questions to Ask

Clarify the random variables, distributional assumptions, independence assumptions, and desired output.

Show enough derivation for the interviewer to follow the reasoning.

Explain how you would validate the result with simulation or sensitivity checks.

What a Strong Answer Covers

A correct setup with definitions, formulas, and boundary conditions.

A step-by-step derivation or estimation plan.

Interpretation of the result, including uncertainty and practical limitations.

Checks for assumptions, edge cases, and numerical stability.

Follow-up Questions

How would the result change if the assumptions were relaxed?

Can you verify the answer with a simulation?

What is the most likely source of estimation error?

Calculate Particle Survival Probability After Time t

Quick Overview

Calculate Particle Survival Probability After Time t

Calculate Particle Survival Probability After Time t

Radioactive-Decay Style Probability

Context

Task

Assumptions

Constraints & Assumptions

Clarifying Questions to Ask

What a Strong Answer Covers

Follow-up Questions

Write your answer

Calculate Particle Survival Probability After Time t

Quick Overview

Calculate Particle Survival Probability After Time t

Calculate Particle Survival Probability After Time t

Radioactive-Decay Style Probability

Context

Task

Assumptions

Constraints & Assumptions

Clarifying Questions to Ask

What a Strong Answer Covers

Follow-up Questions

Write your answer