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Calculate Particle Survival Probability After Time t

Last updated: Mar 29, 2026

Quick Overview

This question evaluates proficiency in probability theory and statistical modeling, focusing on exponential survival functions, independence of events, and discrete counting distributions for multiple identical components.

Upstart

Aug 4, 2025, 10:55 AM

Data Scientist

Technical Screen

Statistics & Math

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}.

Solution

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