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)

$\frac{2}{5}$

Answer 2

Let E = event when each American man is seated
\hspace20mm adjacent to his wife and A = event when Indian man is seated adjacent
\hspace20mm to his wife
Now, \hspace10mm \, \, n(A \cap E)=(4!)\times (2!)^5
Even when each American m an is seated adjacent to his wife.
Again, $\, \, \, \, \, \, \, \, \, \, \, \, \, n(E)=(5!)\times(2!)^4$
$\therefore \, \, \, \, P\bigg(\frac{A}{E}\bigg)=\frac{n(A \cap E)}{n(E)}=\frac{(4!)\times (2!)^5}{(5!)\times (2!)}=\frac{2}{5}$
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 =$\frac{2}{5}$