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Suppose Kn be the number of strings having n X's and n Y's such that in all the prefixes of a string the number of X's is greater than the number of Y's.

Now suppose you are standing at an edge of a swimming pool. You have a bag of n red and n blue balls.Now you are choosing balls randomly from the bag and discarding it. If you pick a red ball you go 1 step forward and if you pick a blue ball you go 1 step backward.

What is the probability that you will not get wet.

A) Kn / (2nCn)

B) n Kn / (2nCn)

C) Kn / (2n)!

d) n kn / 22n

1 Answer

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At the initial stage you are standing at the edge of a swimming pool. Now if you pick a red ball you go 1 step forward and get wet.

Here according to the question you are picking red and blue balls randomly from the bag. Assume you have a copy with you and you are writing X when you are picking a blue ball and Y when you are picking a red ball. 

So, the no. of ways we can pick n red and n blue balls from the bag is equal to the number of strings in the copy = 2nCn .

So, the sample space is 2nCn .

Note that at any time if you have picked more number of red balls than blue balls then you will get wet. In case of strings this will be if any prefix of a string contains more Y's than X's then you will get wet.

So,the cases when you will not get wet is for those strings where any prefix of the string would contain more X's than Y's.

According to the question the number of such strings will be kn .

So, the event space is kn .

So, the probability = kn / 2nCn .

So, option A is the correct answer.

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