Question

An urn contains $6$ blue and $a$ green balls. If the probability of drawing a green ball is double that of drawing a blue ball, then $a$ is equal to:
A: $6$
B: $18$
C: $24$
D: $12$

Answer 1

They have given an urn containing $6$ blue and $a$ green balls. So we can write the total number of balls are$6 + a$. Now, first find the probability of blue balls and then probability of green balls. After finding probabilities we have to equate blue balls to two times of green balls which gives the value of $a$ that is the number of green balls.

Complete Step by Step Solution:
There are $6$ blue and $a$ green balls. Therefore the total number of balls becomes $6 + a$.
Now, to find the value of $a$that is the number of green balls, first we need to find the probability of getting blue balls and then probability of getting green balls.
By using the probability formula, given below we find the required probability.
$P(A) = \dfrac{{n(A)}}{n}$
Where $P(A)$ denotes the probability of A
$n(A)$ is the number of occurrences of A or the number of favorable outcomes.
$n$ is the total number of possible outcomes or the sample space.

Probability of getting blue balls are:
$\Rightarrow P(\text{Blue Ball}) = \dfrac{6}{{6 + a}}$
Probability of getting green balls are:
$\Rightarrow P(\text{Green Ball}) = \dfrac{a}{{6 + a}}$
Now, to find the number of green balls, they have given a condition that the probability of drawing a green ball is double that of drawing a blue ball.
Hence we can write as follows,
$P(\text{Green Ball}) = 2 \times P(\text{Blue Ball})$
$\Rightarrow \dfrac{a}{{6 + a}} = 2 \times \dfrac{6}{{6 + a}}$
In the above equation we can see that the denominator is the same on both the sides, so we can cancel $(6 + a)$ both the sides.
Therefore, we get number of green ball as
$a = 2 \times 6$
$\Rightarrow a = 12$

Hence the value of $a$ is $12$. Therefore the option D is the correct answer.

Note:
In probability questions one thing you need to learn is observation and analysation. When taking the sample space and the number of favorable outcomes we should take care that we are taking the correct value otherwise it will lead to the wrong answer.