Probability 3 Question 11
11. One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that Indian man 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) $\frac{1}{5}$
(2007, 3M)
Show Answer
Solution:
- 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 man is seated adjacent to his wife.
$$ \begin{aligned} & \text { Again, } \quad n(E)=(5 !) \times(2 !)^{4} \\ & \therefore \quad P \frac{A}{E}=\frac{n(A \cap E)}{n(E)}=\frac{(4 !) \times(2 !)^{5}}{(5 !) \times(2 !)^{4}}=\frac{2}{5} \end{aligned} $$
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.
$\therefore \quad$ Required probability $=\frac{2}{5}$
