# Probability of nuclear decay of small staring number of atoms

I came across a rather dubious question that a teacher had put in a power point. It said something like,"Given a sample of 100 atoms of isotope x, after one half life of the said isotope, how many atoms of the original isotope will be left?"

My answer was that it was a trick question because you cannot know exactly how many atoms will be left. Nuclear decay is a random process and it is only possible to make predictions about the probable mass of a macroscopic sample of an isotope that will decay in a given length of time. Is that correct?

Is it actually possible to calculate the probability with such a small number of atoms how many would have decayed after one half life? I don't mind some complicated maths. :)

The probability of N atoms left is $$P(N) = \left( \begin{array}{c} 100 \\ N \end{array} \right) 0.5^{100} \approx \frac{1}{\sqrt{50\pi}}\exp\left(-\frac{(N-50)^2}{50}\right)$$ The expected value will be 50 which is also the most probable value. The standard deviation is 5, so we expect the number of atoms left to be 50 ± 5.