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 put 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

Hint : First of all, we will assume a variable for the probability of Sweety finding the first rose. Then, we will find the probability of Sweety picking the first rose and Shweta picking the first rose. Then we will assume some variable for the number of roses. After that, we will start the game and get an equation of infinite GP. Thus, we can find the number of roses.

Complete step-by-step answer :
Let p be the probability of Sweety picking the first rose.
We know that chances of Sweety picking up the first rose is 3 times more than that of Shweta.
Thus, p = 3(1 – p), where (1 – p) is the probability of Shweta picking up the first rose.
$\Rightarrow$ p = 3 – 3p
$\Rightarrow$ 4p = 3
$\Rightarrow$ p = $\dfrac{3}{4}$ .
Therefore, the probability of Sweety picking up the first rose is $\dfrac{3}{4}$ and Shweta picking up the first rose is $\dfrac{1}{4}$ .
Now, let x be the number of roses in the covered basket. It is given that there are 60 lilies in the covered basket.
Therefore, the total number of flowers in the basket are thus, 60 + x.
And, the probability of picking out a rose will be $\dfrac{x}{60+x}$ .
Let this probability be r. Thus r = $\dfrac{x}{60+x}$
Then, the probability of not getting a rose will be (1 – r).
Now, the probability of Sweety drawing a rose will be she will draw the rose for the first time or she will not draw the rose and Shweta will not draw a rose and then the game resets, as they see the flower and replace it in the basket.
$\Rightarrow$ p = r + ${{\left( 1-\text{r} \right)}^{2}}\text{p}$
$\Rightarrow$ p = r + p + p ${{\text{r}}^{2}}$ ─ 2pr
$\Rightarrow$ p ${{\text{r}}^{2}}$ ─ 2pr + r = 0
But as we know, p = $\dfrac{3}{4}$ .
$\begin{aligned} & \Rightarrow {{r}^{2}}-2\dfrac{3}{4}r+r=0 \\\ & \Rightarrow r\left( 3r-6+4 \right)=0 \\\ & \Rightarrow r\left( 3r-2 \right)=0 \\\ \end{aligned}$
So, r cannot be zero, thus r = $\dfrac{2}{3}$ .
We will r = $\dfrac{2}{3}$ in r = $\dfrac{x}{60+x}$
$\Rightarrow \dfrac{2}{3}=\dfrac{x}{60+x}$
$\Rightarrow$ x = 120
Thus, there are 120 roses in the covered basket.

Note : The probability equation can also be formed in the following way. The probability of Sweety winning is probability of Sweety getting a rose or probability of Sweety not getting a rose and Shweta losing the game, and thus the game resets. $\Rightarrow$ p = r + (1 – r)(1 – p).