# Half-Life for Few Nuclei in a Sample

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.

• Toss 100 coins. Is it guaranteed that you will get exactly 50 heads? Commented Mar 20, 2022 at 7:57

## 1 Answer

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.