Question

A box contains some black balls and 30 white balls. If the probability of drawing a black ball is two-fifths of the probability of drawing a white ball, then find the total number of black balls in the box.

Answer 1

We start solving the problem by assigning the variable for the total number of black balls present inside the box. We then find the probability of drawing white balls and black balls. We then find our first relation using that the probability of drawing a black ball is two-fifths of the probability of drawing a white ball. We then use the fact that the sum of probabilities of drawing white and black balls is 1 to get the probability of drawing white balls. We use this to find the total no. of black balls present in the box.

Complete step-by-step answer:
According to the problem, we have a box that contains some black balls and 30 white balls. We need to find the total number of black balls if the probability of drawing a black ball is two-fifths of the probability of drawing a white ball.
Let us assume the total number of black balls be b. We have 30 white balls and b black balls in the box which makes $\left( 30+b \right)$ balls in the box.
Let us assume the probability of drawing the white and black balls is $P\left( a \right)$ and $P\left( b \right)$.
So, we have $P\left( a \right)=\dfrac{\text{No}\text{. of white balls in bag}}{\text{Total no}\text{. of balls in bag}}$.
$\Rightarrow P\left( a \right)=\dfrac{\text{30}}{\text{30+b}}$ ---(1).
Now, we have $P\left( b \right)=\dfrac{\text{No}\text{. of black balls in bag}}{\text{Total no}\text{. of balls in bag}}$.
$\Rightarrow P\left( b \right)=\dfrac{\text{b}}{\text{30+b}}$ ---(2).
According to the problem, the probability of drawing a black ball is two-fifths of the probability of drawing a white ball.
So, we get $P\left( b \right)=\dfrac{2}{5}\times P\left( a \right)$ ---(3).
We know that the sum of the probabilities of drawing white and black balls is 1.
So, we have $P\left( a \right)+P\left( b \right)=1$.
From equation (3) we get,
$\Rightarrow P\left( a \right)+\dfrac{2P\left( a \right)}{5}=1$.
$\Rightarrow \dfrac{\left( 5+2 \right)P\left( a \right)}{5}=1$.
$\Rightarrow \dfrac{7P\left( a \right)}{5}=1$.
$\Rightarrow P\left( a \right)=\dfrac{5}{7}$.
We substitute this in equation (1).
So, we have $\dfrac{5}{7}=\dfrac{\text{30}}{\text{30+b}}$.
$\Rightarrow \dfrac{1}{7}=\dfrac{\text{6}}{\text{30+b}}$.
$\Rightarrow 30+b=42$.
$\Rightarrow b=12$.
We have found the total number of black balls present in the box as 12.
∴ The total number of black balls present in the box is 12.

Note: We should not make calculation mistakes or confuse with the formulas of finding the probability. We should always know that the total sum of all the probabilities should be 1 and the individual probability is between 0 and 1. We can also solve for the probabilities of drawing the combinations of black and white balls after finding the total no. of black and white balls.