Question

A bag contains 6 white and 4 black balls. 2 balls are drawn at random. Find the probability that they are of the same colour.

Answer 1

Hint: In this question, we are given 10 balls of 2 different colours and we have been asked to find the probability of drawing the balls of the same colour. It is also given that only 2 balls are drawn at once. First, find the number of ways of drawing 2 balls out of 10. Then, find the number of ways of drawing 2 same coloured balls – either 2 white or 2 black. Then put them in the formula of probability and simplify to get your required answer.

Formula used:
1) Probability = ${\displaystyle \frac{\text{Number of favourable items}}{\text{Number of total items}}}$
2) ${}^n{C}_r={\displaystyle \frac{n!}{r!\left(n-r\right)!}}$

Complete step-by-step solution:
We are given 6 white and 4 black balls and we have been asked to find the probability of drawing 2 balls of the same colour.
At first, we will find the number of ways of drawing 2 balls out of 10.
2 balls can be drawn out of 10 balls in the following number of ways –
${\Rightarrow }^{10}{C}_2={\displaystyle \frac{10!}{2!\times \left(10-2\right)!}}$ …. (using ${}^n{C}_r={\displaystyle \frac{n!}{r!\left(n-r\right)!}}$)
${\Rightarrow }^{10}{C}_2={\displaystyle \frac{10\times 9\times 8!}{2!\times 8!}}$
${\Rightarrow }^{10}{C}_2=45$
Therefore, total outcomes are 45.
Now, we will find a number of favourable items.
Favourable items = n(2 white balls appear) + n(2 black balls appear)
Favourable items = ${}^6{C}_2{+}^4{C}_2$
On simplifying we will get,
Favourable items = ${\displaystyle \frac{6!}{2!\times 4!}}+{\displaystyle \frac{4!}{2!\times 2!}}$
Favourable items = ${\displaystyle \frac{6\times 5\times 4!}{2!\times 4!}}+{\displaystyle \frac{4\times 3\times 2!}{2!\times 2!}}$
Favourable items = $15+6=21$
Now, we will put both of them in the formula,
$\Rightarrow$ Probability = ${\displaystyle \frac{\text{Number of favourable items}}{\text{Number of total items}}}$
$\Rightarrow$ Probability = ${\displaystyle \frac{21}{45}}$$={\displaystyle \frac{7}{15}}$

Hence, the probability that the balls are of the same colour is ${\displaystyle \frac{7}{15}}$.

Note: Here, we have to calculate the probability of drawing 2 balls either out of $6$ white or 4 black balls. We used the word ‘or’ in this statement. In such cases, we add the two cases.
In certain cases, we will have to find the probability of drawing 2 balls out of $6$ white and 4 black balls. We used the word ‘and’ in this statement. In such cases, we multiply the two cases.