The setup sounds like a physics problem but it's really just asking you to chain together the exponential survival function and binomial probability.
Start by recognizing that each particle's decay is an independent Bernoulli trial with probability p = e^{-λt} of surviving. Then model the number of survivors as a Binomial(n=100, p) random variable and derive the probability mass function for exactly k and the complement for at least one. Finally, connect the result to the exponential decay law and discuss implications for large n.
Pro tip: Explicitly state the assumptions (independence, identical half-life) and note that the binomial model is exact for any n, but for large n you can approximate with Poisson or normal—this shows you understand both theory and practical scaling.
Given half-life T, the decay constant λ = ln(2)/T. The probability a single particle survives time t is p = e^{-λt} = 2^{-t/T}.
Since particles are independent and identical, the number still undecayed after time t follows a Binomial distribution with parameters n=100 and p as above.
Use the binomial PMF: P(X=k) = C(100, k) * p^k * (1-p)^{100-k}, where C(100,k) is the binomial coefficient.
Use the complement: P(X ≥ 1) = 1 - P(X=0) = 1 - (1-p)^{100}.
Mention that for large n and small p, the Poisson approximation with λ = np can be used, and for large n, the normal approximation applies. Also note the expected number of survivors is np.
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