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Question: A and B are two persons sitting in a circular arrangement with \[8\] other persons. Find the probabi...

A and B are two persons sitting in a circular arrangement with 88 other persons. Find the probability that both AA and B  B\; sit together.

Explanation

Solution

First, we have to define what the terms we need to solve the problem are. The number of people given in this question is 88plus the two others are sitting in a circular arrangement Thus, the total number of people is 1010. We will take the possibility of the two persons sitting on either side of the 88 other persons to interchange their seats.

Complete answer:


Now the number of persons in the given question is 1010( two persons sitting in a circular arrangement with 88 other persons).
So total number of including A,BA,B is 8+2=108 + 2 = 10
Permutation is a way of changing or arranging the elements or objects in a linear order. The formula for permutation for n objects taken r at a time is given by P(n,r) = n!(n−r)!P\left( {n,r} \right){\text{ }} = {\text{ }}\dfrac{{n!}}{{\left( {n - r} \right)!}}
As this a case of circular permutations, ten people randomly be arranged around a circular table in(n−1)!(n - 1)!hence applying the know values in the formula we get (n−1)!=(10−1)!=9!(n - 1)! = (10 - 1)! = 9! ways
Since two particular people are always together, so consider them as one person and let it be saying PP
Now PP with others 88, so 99 can arrange around the table in 8!8!ways (as arranged around a circular table is(n−1)!(n - 1)!) also the two between themselves can arrange in 2!2!ways.
Hence the required probability is 8!×2!9!\dfrac{{8! \times 2!}}{{9!}} 8!×2!9×8!=29\dfrac{{8! \times 2!}}{{9 \times 8!}} = \dfrac{2}{9}(since 9!9! can be written as 9×8!9 \times 8! and canceling each other)
Therefore, the probability for both AA and B  B\; sit together is 29\dfrac{2}{9}

Note: For probability first, determine a single event with a single outcome and identify the total number of outcomes that can occur then finally divide the number of events by the number of possible outcomes.
Which is Probability= number of favourable outcomestotal number of outcomeProbability = \dfrac{{{\text{ }}number{\text{ }}of{\text{ }}favourable{\text{ }}outcomes}}{{total{\text{ }}number{\text{ }}of{\text{ }}outcome}}.