Question

One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that Indian m an is seated adjacent to his wife given that each American man is seated adjacent to his wife, is

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{2}{5}$
• D. $\frac{1}{5}$

Answer 1

Answer: (C)

Let $E=$ event when each American man is seated adjacent to his wife
and $A=$ event when Indian man is seated adjacent to his wife
Now, $n(A\cap E)=(4!)\times (2!{)}^5$
Even when each American m an is seated adjacent to his wife.
Again, $n(E)=(5!)\times (2!{)}^4$
$\therefore P\left(\frac{A}{E}\right)=\frac{n(A\cap E)}{n(E)}=\frac{(4!)\times (2!{)}^5}{(5!)\times (2!)}=\frac{2}{5}$
Alternate Solution
Fixing four American couples and one Indian man in between any two couples; we have $5$ different ways in which his wife can be seated, of which $2$ cases are favourable.
$\therefore$ Required probability =$\frac{2}{5}$