Question

A plastic bag contains $12$ red marbles, $15$ green marbles and $5$yellow marbles. Find the number of additional red marbles that must be added to the bag so that the probability drawing the red marble is $\dfrac{3}{5}$?
A. $13$
B. \[18\]
C. $28$
D. $32$
E. $40$

Answer 1

Hint:
If we have $x$ number of red marbles, $y$ number of green marbles and $z$ number of yellow marbles then there will be total of $x + y + z$ marbles in the bag and the probability of picking out of the red marbles will be $\dfrac{x}{{x + y + z}}$.

Complete step by step solution:
According to the solution, the plastic bag contains $12$ red marbles, $15$ green marbles and $5$ yellow marbles. Therefore the bag has three different types of balls but now if we add some red marbles in it then the probability of drawing the red marble will become $\dfrac{3}{5}$
So basically the probability of drawing red marble$ = \dfrac{{{\text{number of red marbles}}}}{{{\text{total number of marbles in bag}}}}$
As we know the number of each type of the marble so we get that
Total number of marbles $ = 12 + 15 + 5 = 32$
So basically now the probability of drawing red ball $ = \dfrac{{12}}{{32}}$
Initially the probability is $ = \dfrac{{12}}{{32}}$ but after the addition of the red more marbles the probability is said to be $\dfrac{3}{5}$
So let us suppose that we add $x$ more red marbles into the bag
So total number of marbles now are $ = (12 + x) + 15 + 5 = 32 + x$
Total number of red marbles now $ = 12 + x$
But now we are aware of the probability of drawing the red ball which is given as $\dfrac{3}{5}$
So we can use the formula of the probability to get the value of the additional number of red balls.
Probability of drawing red marble$ = \dfrac{{{\text{number of red marbles}}}}{{{\text{total number of marbles in bag}}}}$
\[
  \dfrac{3}{5} = \dfrac{{12 + x}}{{32 + x}} \\
  3(32 + x) = 5(12 + x) \\ \Rightarrow
  96 + 3x = 60 + 5x \\ \Rightarrow
  5x - 3x = 96 - 60 \\ \Rightarrow
  2x = 36 \\ \Rightarrow
  x = 18 \\
 \]

So we need to add $18$ more red balls into the bag so that the probability of drawing the red ball becomes $\dfrac{3}{5}$

Note:
We should know that the probability of any event always lies between the value from 0 to 1.
The probability is 1 if it has $100\% $ chance of being done and is 0 if it has 0% chance of being done.