Question

A bag contains 20 silver balls and x iron balls. If one ball is drawn from the bag, what is the probability that it is silver?

Answer 1

Hint: Use permutations and combinations to get the number of favourable outcomes and the total number of outcomes. Think of the basic interpretation of combination.

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 $={\displaystyle \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}}$ .

Now, let’s move to the solution to the above question.

We are given that the number of silver balls in the bag is 20 and the number of iron balls in the bag is x.

First, let us try to find the number of favourable outcomes. Whenever we select one out of the twenty silver balls, it is counted as a favourable event. We can mathematically represent this as:

Ways of selecting one out of twenty silver balls = ${}^{20}{C}_1$ .

Now let us try to calculate the total number of possible outcomes. It is counted as one possible outcome whenever we draw a ball, whether it is silver or an iron ball.

Now let us try to represent it mathematically.

Ways of selecting one out of ( 20 + x) balls present in the bag = ${}^{20+x}{C}_1$ .

Now, using the above results let us try to find the probability of drawing silver balls:

$\text{Probability}={\displaystyle \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}}$

$$\Rightarrow \text{Probability}={\displaystyle \frac{{}^{20}{C}_1}{{}^{20+x}{C}_1}}$$

Now, using the formula ${}^n{C}_r={\displaystyle \frac{n!}{r!(n-r)!}}$ , we get:

$$\text{Probability}={\displaystyle \frac{{\displaystyle \frac{20!}{(20-1)!1!}}}{{\displaystyle \frac{(20+x)!}{(20+x-1)!1!}}}}$$

$$\Rightarrow \text{Probability}={\displaystyle \frac{{\displaystyle \frac{20!}{19!}}}{{\displaystyle \frac{(20+x)!}{(19+x)!}}}}$$

We know n! can be written as n(n-1)! . So, our equation becomes:

$$\text{Probability}={\displaystyle \frac{{\displaystyle \frac{20\times 19!}{19!}}}{{\displaystyle \frac{(20+x)(19+x)!}{(19+x)!}}}}$$


$$\therefore \text{Probability}={\displaystyle \frac{20}{20+x}}$$

So, the probability of drawing a silver ball from a bag of 20 silver balls and x iron balls is ${\displaystyle \frac{20}{20+x}}$ .


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.