# Bayesian inference problem involving Poisson statistics

In a laboratory a set of samples of radioactive materials is kept. Half of the samples consists of N atoms with a half life of t1, and the other half consists of N atoms with a half life of t2. A sample is drawn randomly from the entire set, during a time interval of T, no decays are detected. Compute the probability that the sample consists of atoms with a half life of t1. N is assumed to be so large that one can ignore the decrease in decay rate during the time interval of T.

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#### Solution Preview

If the half life of atoms is t, then with N atoms at time T = 0, the number of atoms as a function of time is given as:

N(T) = N 2^(T/t)

The rate R at which the decay happens at T = 0 is the derivative at T = 0. We have:

dN/dT = N 2^(T/t) ...

#### Solution Summary

A detailed solution is given for the example question about probability using Poisson statistics. Familiarity with Bayes' theorem is assumed.