Question

There are $2$ red and $2$ yellow flowers in a basket. A child picks up at random three flowers. What is the probability of picking up both the yellow flowers?

Answer 1

It is given that there are $2$ red and $2$ yellow flowers in a basket which means the total number of flowers are $4$.

The number of ways of picking three flowers is:

${}^4{C}_3=\frac{4!}{(4-3)!\times 3!}=\frac{4!}{1!\times 3!}=\frac{4\times 3\times 2\times 1}{1\times 3\times 2\times 1}=4$

Therefore, $n\left(S\right)=4$

Let $A$ denote the event of picking three flowers out of which two are yellow, therefore,

$n\left(A\right){=}^2{C}_2{\times }^2{C}_1=\frac{2!}{(2-2)!\times 2!}\times \frac{2!}{(2-1)!\times 1!}=\frac{2!}{0!\times 2!}\times \frac{2!}{1!\times 1!}=\frac{2\times 1}{1\times 2\times 1}\times \frac{2\times 1}{1\times 1}=2$

Thus, probability of picking three flowers out of which two are yellow is:

$P\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)}=\frac{2}{4}=\frac{1}{2}$

Hence, probability of picking up both the yellow flowers is $\frac{1}{2}$.