Question

A bag contains 10 white balls and X black balls. If the probability of drawing white balls is double that of black balls. What is the probability of drawing a black ball?

Answer 1

Hint: For finding the probability of drawing white ball, the number of favourable cases is equal to number of ways of selecting 1 ball out of the 10 white balls, while for finding the probability of drawing a black ball is equal to the number ways of selecting one out of the x black balls. The total number of possible outcomes is equal to the number of ways of selecting one out of (10+x) balls. SO, get the probabilities and equate probability of drawing white balls with twice the probability of drawing black balls, to get x and hence, the required result.

Complete step-by-step answer:
Before moving to the question, let us talk about probability.
Probability in simple words is the possibility of an event to occur.
Probability can be mathematically defined as $=\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$ .
Now, let’s move to the solution to the above question.
Given:
Number of white balls in the bag = 10
Number of black balls in the bag = X
Let us try to find the number of favourable outcomes for drawing white ball:
So, whenever we select one out of the 10 white balls, it is counted as a favourable event.
We can mathematically represent this as:
Ways of selecting one out of 10 white balls = $^{10}{{C}_{1}}$ .
Similarly, we can say that the favourable outcomes for drawing black balls is equal to the number of ways of selecting one out of X black balls, i.e., $^{X}{{C}_{1}}$ .
Now let us try to calculate the total number of possible outcomes.
So, it is counted as one of the possible outcomes whenever we draw a ball, whether it is white or a black ball.
Now let us try to represent it mathematically.
We get;
Ways of selecting one out of ( 10 + x) balls present in the bag = $^{10+x}{{C}_{1}}$ .
Now, using the above results let us try to find the probability of drawing white balls:

$\text{Probability}=\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$
\[\Rightarrow \text{Probability}=\dfrac{^{10}{{C}_{1}}}{^{10+X}{{C}_{1}}}\]
Similarly we can say probability of drawing black ball is $\dfrac{^{X}{{C}_{1}}}{^{10+X}{{C}_{1}}}$
Now as it is given that probability of drawing white balls is double that of black balls, we can say:
$\dfrac{^{10}{{C}_{1}}}{^{10+X}{{C}_{1}}}=\dfrac{2{{\times }^{X}}{{C}_{1}}}{^{10+X}{{C}_{1}}}$
${{\Rightarrow }^{10}}{{C}_{1}}=2{{\times }^{X}}{{C}_{1}}$
$\Rightarrow 10=2\times X$
$\Rightarrow X=5$
Therefore, the probability of drawing black ball is: $\dfrac{^{X}{{C}_{1}}}{^{10+X}{{C}_{1}}}=\dfrac{^{5}{{C}_{1}}}{^{10+5}{{C}_{1}}}=\dfrac{5}{15}=\dfrac{1}{3}$ .

Note: It is preferred that while solving a question related to probability, always cross-check the possibilities, as there is a high chance you might miss some or have included some extra or repeated outcomes. Also, when a large number of outcomes are to be analysed then permutations and combinations play a very important role as we see in the above solution.