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Question: A plastic bag contains \(12\) red marbles, \(15\) green marbles and \(5\)yellow marbles. Find the nu...

A plastic bag contains 1212 red marbles, 1515 green marbles and 55yellow marbles. Find the number of additional red marbles that must be added to the bag so that the probability drawing the red marble is 35\dfrac{3}{5}?
A. 1313
B. 1818
C. 2828
D. 3232
E. 4040

Explanation

Solution

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

Complete step by step solution:
According to the solution, the plastic bag contains 1212 red marbles, 1515 green marbles and 55 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 35\dfrac{3}{5}
So basically the probability of drawing red marble=number of red marblestotal number of marbles in bag = \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 = 12 + 15 + 5 = 32
So basically now the probability of drawing red ball =1232 = \dfrac{{12}}{{32}}
Initially the probability is =1232 = \dfrac{{12}}{{32}} but after the addition of the red more marbles the probability is said to be 35\dfrac{3}{5}
So let us suppose that we add xx more red marbles into the bag
So total number of marbles now are =(12+x)+15+5=32+x = (12 + x) + 15 + 5 = 32 + x
Total number of red marbles now =12+x = 12 + x
But now we are aware of the probability of drawing the red ball which is given as 35\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=number of red marblestotal number of marbles in bag = \dfrac{{{\text{number of red marbles}}}}{{{\text{total number of marbles in bag}}}}

35=12+x32+x 3(32+x)=5(12+x) ⇒96+3x=60+5x ⇒5x−3x=96−60 ⇒2x=36 ⇒x=18  \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 1818 more red balls into the bag so that the probability of drawing the red ball becomes 35\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%100\% chance of being done and is 0 if it has 0% chance of being done.