I read somewhere in some textbook that

"The concept of half life and mean half life becom meaningless ehen you have a sample containing only a few nuclei. For example, if you are given say, 100 active nuclei on a sample holder, and the half life of the material is one hour, then it does not mean that you will be left with 50 nuclei after 1 hour! As the number of nuclei in the sample is increased-though we can not say which nuclei will decay at a given instant- we can say that about half of them will decay after one half life".

I didn't get the thing why one can't discuss half life concept for few number of nuclei. 2nd is the example taken here I didn't understand it what does it trying to convey to readers that after 1 half life 50 wont decay.

  • 2
    $\begingroup$ Toss 100 coins. Is it guaranteed that you will get exactly 50 heads? $\endgroup$
    – PM 2Ring
    Commented Mar 20, 2022 at 7:57

1 Answer 1


Consider tossing a coin $2$ times which is equivalent to observing 2 unstable nuclei in a time interval equal to one half life.
What is the probability that there will be $0,\,1,\,2$ heads (nucleus decays).
The answer is $0.25,0.5,0.25$ consisting of $\rm TT,\,HT,\, TH,\,HH$ so the chance of just one head (decay) is $0.5$.
So far so good!

Now what about $10$ coin tosses (unstable nuclei)?
The probability of four or less heads is $\frac{193}{512}$ which is the same as the probability of six or more heads whilst the probability of exactly five heads is $\frac {63}{256}$.
So the probability of exactly $5$ of the $10$ unstable nuclei decaying in a time of one half life is $\frac {63}{256}$.

For your example of $100$ unstable nuclei the probability of exactly $50$ decaying in one half life is $0.07959$

As the number of nuclei increases the probability of exactly half of them decaying gets smaller so for $10,000$ unstable nuclei the probability of exactly $5,000$ decaying is $0.007979$.

The distribution of such events is called a binomial and I used the WolframAlpha binomial distribution calculator to crunch the numbers.


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