Question

A purse contains n coins of unknown value; a coin drawn at random is found to be a sovereign; what is the chance that it is the only sovereign in the bag?

Answer 1

For a bag containing $n$ coins,

P( drawing 1 sovereign out 1 sovereign)$=\frac{1}{n}$

P( drawing 1 sovereign out 2 sovereigns)$=\frac{2}{n}$

P( drawing 1 sovereign out 3 sovereigns)$=\frac{3}{n}$ and so on

So the total probability is $\frac{1}{n}+\frac{2}{n}+\cdot \cdot \cdot \frac{n}{n}=\frac{n(n+1)}{2n}=\frac{n+1}{2}$

P( drawing 1 sovereign out 1 sovereign)$=\frac{1}{n}$

Hence, probability is $\frac{\frac{1}{n}}{\frac{n+1}{2}}=\frac{2}{n(n+1)}$