An interesting probability!

Let describe a process AA as follows:

Suppose you have a bag, and it have a ball which either have red\red{red} or blue\blue{blue} color (you don't know what color it have). You put a red\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 AA ''nn'' times and every time you get red\red{red} ball as output then the probability that the main ball of the whole process is blue\blue{blue} is 1002n+1%\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{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{blue} color)

In this scenario, each time you will be getting either of the two redred 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 nn times? You can prove it yourself, is 2n2^n

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

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

\therefore probability that the ball left inside the bag after applying the process nn times =12n+1=\dfrac{1}{2^n+1}

For percentage multiply it by 100100 getting the probability percentage that the main ball after applying the process nn times =1002n+1%=\dfrac{100}{2^n+1}\%


Note :

  • Inspiration

  • Meaning of doing process AA ''nn'' 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 ''nn'' times and after doing it nthn_{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

Note by Zakir Husain
4 months, 3 weeks ago

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@Jeff Giff- See if you find this interesting enough to add in RedMaths\large{RedMaths}

Zakir Husain - 4 months, 3 weeks ago

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Interesting @Zakir Husain

Kriti Kamal - 4 months, 3 weeks ago

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