Question

A box contains 24 balls of which some are yellow in color. If 12 more yellow balls are put in the box and a ball is drawn at random, the probability of drawing a yellow ball doubles that what it was before. Find the number of yellow balls in the box initially.

Answer 1

Hint: We solve this problem by assuming the number of yellow balls as some variable.
We use the formula of probability as
\[P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]
We find the probability in two cases then we use the given condition to find the required value.

Complete step by step answer:
We are given that there are some yellow balls in the box of 24 balls.
Let us assume that the number of yellow balls as \[x\]
Now, let us assume that the probability of getting a yellow ball from a box of 24 balls as \[{{P}_{1}}\]
We know that the formula of probability as
\[P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]
By using the above formula we get the probability of getting a yellow ball initially as
\[\Rightarrow {{P}_{1}}=\dfrac{x}{24}\]
We are given that 12 more yellow balls are added to the box.
Here we can see that the total number of balls is \[24+12=36\] and the number of yellow balls is \[x+12\]
Let us assume that the probability of getting a yellow ball now as \[{{P}_{2}}\]
By using the probability formula we get the probability of getting a yellow ball after adding 12 balls as
\[\Rightarrow {{P}_{2}}=\dfrac{x+12}{36}\]
We are given that the probability of drawing a yellow ball after adding the balls doubles that what it was before.
By converting the above statement into mathematical equation we get
\[\Rightarrow {{P}_{2}}=2{{P}_{1}}\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow \dfrac{x+12}{36}=2\times \dfrac{x}{24} \\
 & \Rightarrow \dfrac{x+12}{3}=x \\
\end{align}\]
By cross multiplying the terms in the above equation we get
\[\begin{align}
  & \Rightarrow x+12=3x \\
 & \Rightarrow 2x=12 \\
 & \Rightarrow x=4 \\
\end{align}\]
Therefore we can conclude that there are 4 yellow balls initially in the box.

Note:
 Students may make mistakes in taking the statement into a mathematical equation.
We are given that the probability of drawing a yellow ball after adding the balls doubles that what it was before.
By converting the above statement into a mathematical equation we get
\[\Rightarrow {{P}_{2}}=2{{P}_{1}}\]
But students may do mistake in understanding the statement that is they assume in the reverse order and take that equation as
\[\Rightarrow 2{{P}_{2}}={{P}_{1}}\]
This gives the wrong answer because the final probability is twice the initial probability.
So, we need to understand the given statement correctly.