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3. Pseudo-random numbers

In-class exercises

1. One isotope of thallium (${}^{208}Ti$) has a half-life $t_{1/2}$ =3.053 minutes, decaying to ${}^{208}$Pb. You will write a simple for loop using pseudo-random numbers to track the populations of ${}^{208}$Tl and ${}^{208}$Pb in a sample over 1,000 seconds, starting with 1,000 atoms of ${}^{208}$Tl and 0 atoms of ${}^{208}$Pb.

The probability of decay in a time interval dt is $p=1-e^{-dt/\lambda }$ where $\lambda =t_{1/2}$/log2. Set dt=1, and at each step, evaluate the probability that each ${}^{208}$Tl atom decays by creating an array of random numbers from 0 to 1 whose length is the number of ${}^{208}$Tl atoms (use rng.random). For each atom, it will decay if the random number generated, $n<p$. You’ll need to have two arrays to store the populations for Tl and Pb atoms at each step. (Use np.where as described in the notes.)

At each timestep, sum over the number of atoms that do decay (using np.sum on the result of the np.where call), and use this number to update the total number of thallium and lead atoms. Make a plot of the number of atoms of each species against time.