Question

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

• A. $1/2$
• B. $1/3$
• C. $2/5$
• D. $1/5$

Answer 1

Answer: (C)

$2/5$

Let E be the event where an American is seated adjacent to his wife.
A be the event where an Indian is seated adjacent to his wife.

Now
$n(A\cap E)=4!\times (2!{)}^5$ (Each couple is taken together as one entity, so we have total 5 things to be arranged in a circle, which can be done in $(5-1)!=4!$ number of ways. Also, each couple as one entity can be seated in 2 ways.)

$n\left(E\right)=5!\times (2!{)}^4$ (Each American couple is taken as one entity, and an Indian man and woman separately. Hence, the arrangement of 6 things in a circle =5! Also each American couple as one entity can be seated in 2 ways. )

$P\left(\frac{A}{E}\right)=\frac{n(A\cap E)}{n\left(E\right)}=\frac{4!\times 32}{5!\times 16}$
$=\frac{2}{5}$
Hence required probability is equal to $\frac{2}{5}$.