Question

A covered basket of flowers has some lilies and roses. In search of rose, Sweety and Shweta alternately pick up a flower from the basket but puts it back if it is not a rose. Sweety is $3$ times more likely to be the first one to pick a rose. If sweety begin this 'rose hunt' and if there are $60$ lilies in the basket, find the number of roses in the basket.

Answer 1

Let $p$ be the probability of Sweety getting the first rose, and $r$ be the probability of getting a rose.

So, $p=3(1-p)\Rightarrow p=\frac{3}{4}$ (where $(1-p)$ is the probability of Shweta getting a rose)

and $r=\frac{x}{60+x}$ where $x$ is the number od roses.

ie, $p=r+(1-r{)}^{2}p$ ($\because (1-r{)}^2$ shows that none of them got a rose in 1st try which rounds back again to get another 1st try)

$0=r-2pr+r^2\Rightarrow r=0orr=\frac{1}{2}$

As $r\ne 0$,

$\frac{1}{2}=\frac{x}{60+x}\Rightarrow Thereare60Roses.$