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.