Question

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 :

• A. $\frac{3}{10}$
• B. $\frac{2}{5}$
• C. $\frac{1}{5}$
• D. $\frac{3}{4}$

Answer 1

The Correct Option is B

$E_{1}:$ Event that first ball drawn is red.
$E_{2}:$ Event that first ball drawn is black.
$E$ : Event that second ball drawn is red.
$P(E)=P\left(E_{1}\right) \cdot P\left(\frac{E}{E_{1}}\right)+P\left(E_{2}\right) \cdot P\left(\frac{E}{E_{2}}\right)$
$=\frac{4}{10} \times \frac{6}{12}+\frac{6}{10} \times \frac{4}{12}=\frac{2}{5}$

Answer 2

Let's handle the problem with two different cases:

Case 1:

When we draw a black ball first, we will find the probability of the black ball. But why so?

This is because the probability of finding the red ball the second time depends on what we take out the first time.

 

⇒P(black ball) = \(\frac{6}{(4+6)}\) = \(\frac{3}{5}\)  —--------(1)

 

As stated in the problem, we will add two more balls of the color we took which means that we will add two more black balls.

The total number of black balls now =8

Now the probability of getting the red ball = \(\frac{4}{(4+8)}\)= \(\frac{1}{3}\)  —--------(2)

Case 1 is now completed.

Case 2:

In order to find the total probability of getting the red ball from the first case is by multiply (1) and (2) because one leads to another.

⇒P(red ball) = \(\frac{3}{5}\)×\(\frac{1}{3}\) = \(\frac{1}{5}\) 

Similarly, we will make the second case:

⇒P(red ball) = \(\frac{4}{(4+6)}\) = \(\frac{2}{5}\)  

Now we will add two more red balls to it and then we will get a total of six red balls.

⇒P(red ball) = \(\frac{6}{(6+6)}\) = \(\frac{1}{2}\)  

Therefore in the second case,

⇒P(red ball) =  \(\frac{2}{5}\)×\(\frac{1}{2}\) = \(\frac{1}{5}\)

Now we will add both the cases 

⇒P(red ball) = \(\frac{1}{5}\) + \(\frac{1}{5}\) =\(\frac{2}{5}\)

Hence option D is the correct answer.