An interesting probability!

Let describe a process $A$ as follows:

Suppose you have a bag, and it have a ball which either have $\red{red}$ or $\blue{blue}$ color (you don't know what color it have). You put a $\red{red}$ ball inside that bag and pick a ball at random from the bag, let the color of the ball you got be called the output of the process and the ball left inside the bag as the main ball of the process.

Now suppose you do process $A$ ''$n$'' times and every time you get $\red{red}$ ball as output then the probability that the main ball of the whole process is $\blue{blue}$ is $\dfrac{100}{2^n+1}\%$

Proof :

Let us analyze each scenario one by one

First scenario (The ball in the start was of $\blue{blue}$ color)

In this case there is only one and only one output coming every time , that is the red ball you put inside the bag in starting is the same ball you are getting as output each time you does the process

Second scenario (The ball in the start was of $\blue{blue}$ color)

In this scenario, each time you will be getting either of the two $red$ ball as an output.

So in total how many different ways can you pick ball from the bag and then put it in the bag and then again pick a ball, and does this $n$ times? You can prove it yourself, is $2^n$

So total number of possible events $=2^n+1$

Number of possible events in which there is $\blue{blue}$ ball left$=1$

$\therefore$ probability that the ball left inside the bag after applying the process $n$ times $=\dfrac{1}{2^n+1}$

For percentage multiply it by $100$ getting the probability percentage that the main ball after applying the process $n$ times $=\dfrac{100}{2^n+1}\%$

Note :

• Inspiration

• Meaning of doing process $A$ ''$n$'' times : It means that you put the red ball inside the bag and then pick a ball randomly, after that you put the ball back into the bag and then again pick a ball randomly and does this continuously ''$n$'' times and after doing it $n_{th}$ time you don't put the output ball back in the bag this time, and the other ball left in the bag is the main ball of the whole process