Question

Thirteen persons take their places at a round table, show that it is five to one against two particular persons sitting together.

Answer 1

Hint: Here we will find the probabilities of taking 2 persons around a round table from 13 persons and then we will find the probability of that two person not sitting together. Then we just need to take the ratio.

Complete step-by-step answer:
It is given that thirteen persons take their places at a round table and we have to prove that it is 5 to 1 against two particular persons.
So basically we need to prove $ \dfrac{{P\left( {\overline A } \right)}}{{P\left( A \right)}} = 5:1 $
Now we know that for round table Arrangement. If there are n persons, then Possible arrangement is $ (n - 1)! $ .So we have 13 persons and hence for 13 persons is $ (13 - 1)! = 12! $ .
Now for particular two persons, let us consider them one and hence, we now have 12 persons, so now for twelve persons is $ (12 - 1)! = 11! $
So expected outcome= $ 11! \times 2! $ (Arranging them together)
Therefore the total outcome is $ 12! $ .
Hence Probability of a person sitting together is $ \dfrac{{11! \times 2!}}{{12!}} = \dfrac{1}{6} $ .
Therefore $ P\left( {\overline A } \right) = 1 - \dfrac{1}{6} = \dfrac{5}{6} $ .
Therefore, Ratio= $ \dfrac{{P\left( {\overline A } \right)}}{{P\left( A \right)}} = \dfrac{{\dfrac{5}{6}}}{{\dfrac{1}{6}}} = 5:1 $

Note: So in this type of question first of all we have to find the possible arrangement and then we have to find $ P(A) $ and $ P\left( {\overline A } \right) $ and then on putting their value we can find the ratio.