A bag contains $4$ red and $6$ black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is

The correct option is D

$\frac{2}{5}$

The explanation for the correct answer:

Step 1: Probability that it is black at first and red at second.

Total number of black balls = $6$

Total number of red balls = $4$

Probability of a black ball in the first draw = $\frac{6}{10}$

If a black ball is drawn first, then two black balls are added to the bag.

Now, the number of black balls = $8$ and

Total number of balls = 12

Thus probability that it is a red ball in the second draw = $\frac{4}{12}$

Step 2: Probability of drawing a red ball at both times.

Probability of a red ball in the first draw = $\frac{4}{10}$

If a red ball is drawn first, then two red balls are added to the bag.

Now, the number of red balls = $6$

Total number of balls = $12$

Thus probability that it is a red ball in the second draw = $\frac{6}{12}$

Step 3: Total probability of drawing a red ball.

$$\begin{gathered} =\left(\frac{6}{10}\times \frac{4}{12}\right)+\left(\frac{4}{10}\times \frac{6}{12}\right) \\ =\frac{24}{120}+\frac{24}{120} \\ =\frac{48}{120} \\ =\frac{2}{5} \end{gathered}$$

Hence, option (D) is the correct answer.