Probability
Conditional Probability and Permutations
Grade 12
Question:
<p>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</p>
<p>(a) \(\frac{1}{2}\)</p>
<p>(b) \(\frac{1}{3}\)</p>
<p>(c) \(\frac{2}{5}\)</p>
<p>(d) \(\frac{1}{5}\)</p>
Step-by-Step Solution
Key Concept: Use circular permutations and condition on the American couples sitting together. Calculate the probability by counting favorable arrangements.
<p>We have 1 Indian couple and 4 American couples seated around a circular table (10 people total). Given that each American man is adjacent to his wife, we need to find the probability that the Indian man is adjacent to his wife. When each American couple sits together as a unit, we have 5 units to arrange around a circle: $(5-1)! = 24$ ways. Within each American couple, there are 2 arrangements (man-woman or woman-man), giving $2^4 = 16$ arrangements. The Indian couple can be arranged in 2 ways. Among these 24 circular arrangements of 5 units, the Indian man's wife can be in one of 2 adjacent positions or 3 non-adjacent positions relative to him (when considering the circular arrangement). The conditional probability is $\frac{1}{3}$.</p>
Correct Answer: B