Question

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different color is

Answer 1

Hint: - Here we have to go through by the formula of finding the probability by finding the total number of outcomes and by calculation the favorable outcomes.

Complete step-by-step answer:
Here in the question it is given that an urn contains nine balls of which three are red, four are blue and two are green.
First of all we have to calculate total number of ball in a bag ${\text{ = 4B + 3R + 2G = 9}}$balls
Here B=blue balls, ${\text{R = }}$red balls and ${\text{G = }}$green balls
Now we have to find out the total number of outcomes when the three balls are drawn at random without replacement from the urn.
As we know that the formula of choosing r objects from n objects is${}^n{C_r}$.
By this formula we can say that the number of ways for choosing 3 balls out of 9 can be${}^9{C_3}$.
$\therefore$Total number of outcome=${}^9{C_3}$
Now we have to find out the favorable outcome in which all the three balls have different colors.
So for choosing all balls different in color we have to choose one red ball. One blue and one green ball from their numbers.
I.e. choosing one red ball from 3 red balls=${}^3{C_1}$,
Choosing one blue ball from 4 blue balls=${}^4{C_1}$,
And choosing one green ball from 2 green balls=${}^2{C_1}$,
Now the total number of favorable outcome is ${}^3{C_1} \times {}^4{C_1} \times {}^2{C_1}$
$\therefore P{\text{ = }}\dfrac{{{\text{favorable outcome}}}}{{{\text{total outcome}}}} = \dfrac{{{}^3{C_1} \times {}^4{C_1} \times {}^2{C_1}}}{{{}^9{C_3}}} = \dfrac{{3 \times 4 \times 2}}{{\dfrac{{9!}}{{(9 - 3)! \times 3!}}}} = \dfrac{{24 \times 3!}}{{9!}} \times 6! = \dfrac{2}{7}$
As we know that ${}^n{C_r}$ can be written as $\dfrac{{n!}}{{(n - r)! \times r!}}$ and we simply write ${}^n{C_1}$ as 1.
Hence the required probability of getting three balls of a different color is$\dfrac{2}{7}$.

Note: - Whenever we face such type of question the key concept for solving the question is to find the probability you have to always calculate the favorable outcome and total number of outcome to find the probability of given statement and always remember how to choose r numbers from n number it is very helpful in terms of finding favorable outcomes and total number of outcomes.