Calculate Particle Survival Probability After Time t
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.
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Aug 4, 2025, 10:55 AM
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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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