Question

A bag contains white, black and red balls only. A ball is drawn random from the bag. If the probability of getting a white ball is $\dfrac{{\text{3}}}{{{\text{10}}}}$ and that of a black ball is $\dfrac{{\text{2}}}{{\text{5}}}$​, then find the probability of getting a red ball. If the bag contains ${\text{20}}$ black balls, then find the total number of balls in the bag.

Answer 1

Given the probability of drawing a ball of particular colour. We have to find the total number of balls present in the bag. First, we will find the probability of drawing a red ball. Then, we will find the probability of drawing a black ball using the probability formula and equate the probability given in the question with the calculated probability. Then, we will substitute the values in the equation and simplify the equation to determine the total number of balls.

Formula used:
$P\left( E \right) = \dfrac{{{\text{Number of outcomes for event E}}}}{{{\text{Total number of outcomes}}}}$

Complete step-by-step answer:
We are given the probability of drawing a ball of white and black colour. Now, we will find the probability of drawing a red ball by subtracting the sum of the probabilities of drawing a white and black ball from one.
$\Rightarrow P\left( {{\text{Red ball}}} \right) = 1 - \left( {\dfrac{{\text{3}}}{{{\text{10}}}} + \dfrac{{\text{2}}}{{\text{5}}}} \right)$
$\Rightarrow P\left( {{\text{Red ball}}} \right) = 1 - \left( {\dfrac{{3 + 4}}{{{\text{10}}}}} \right)$
$\Rightarrow P\left( {{\text{Red ball}}} \right) = 1 - \left( {\dfrac{7}{{{\text{10}}}}} \right)$
On simplifying the expression, we get:
$\Rightarrow P\left( {{\text{Red ball}}} \right) = \dfrac{{10 - 7}}{{{\text{10}}}}$
$\Rightarrow P\left( {{\text{Red ball}}} \right) = \dfrac{3}{{{\text{10}}}}$
Thus, the probability of drawing a red ball is $\dfrac{3}{{{\text{10}}}}$
Now, compute the probability of drawing a black ball.
$P\left( {black} \right) = \dfrac{{{\text{Number of black balls in a bag}}}}{{{\text{Total number of balls in a bag}}}}$
Let the total number of balls in a bag be $n$.
We will substitute ${\text{20}}$ for the number of black balls in a bag and $\dfrac{{\text{2}}}{{\text{5}}}$ for probability of drawing a black ball into the expression.
$\Rightarrow \dfrac{2}{5} = \dfrac{{{\text{20}}}}{n}$
Cross multiply the terms, we get:
$\Rightarrow 2n = 20 \times 5$
On further simplifying the expression, we get:
$\Rightarrow n = \dfrac{{100}}{2}$
$\Rightarrow n = 50$

Final answer: Therefore, the total number of balls in a bag are $50$.

Note:
In such types of questions students mainly make mistakes while choosing the formula that must be applied to get the answer. Students can make mistakes while finding the value of the probability of drawing black colour ball from the bag.