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

1. Derive the probability that exactly k particles (out of 100) remain undecayed after time t.
2. 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}.