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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. :)

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The probability of N atoms left is \begin{equation} 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) \end{equation} 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.

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  • $\begingroup$ Per Arve - sorry I am not quite sure how that calculation works. Say for example I wanted to know what the probability would be of there being exactly 50 atoms left, how would I calculate that? In the equation you have given it looks like one is multiplying some by .5 to the power of 100 which I get the feeling will come out as a very tiny number. $\endgroup$ – Peter Hunt Jan 23 '16 at 17:42
  • $\begingroup$ Peter Hunt - I have edited my answer. Hope it is clearer now. $\endgroup$ – Per Arve Jan 23 '16 at 18:23
  • $\begingroup$ Thanks Per Arve. I tried a binomial probability calculator and found that there was something like a 72% probability of there being between 45 and 55 atoms left out of a sample of 100. The probability that there were exactly 50 atoms left was about 8%. That sounds about right to me. $\endgroup$ – Peter Hunt Jan 23 '16 at 20:32
  • $\begingroup$ I just tried your calculation and got exactly the same answer. When N = 50 , the exp term becomes exp zero = 1, leaving just the 1 over root 50 pi = 0.079 In others words approximately 8% . $\endgroup$ – Peter Hunt Jan 23 '16 at 20:43

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