Thirty-two persons X_1, X_2, ldots, X_{32} ar | vedclass

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(C) There are $32$ persons seated around a circular table. Fix the position of $X_1$. There are $31$ remaining seats where $X_3$ can be placed,each wi

Vedclass · Accepted Answer

$\frac{8}{31}$

Vedclass · Answer

(C) There are $32$ persons seated around a circular table. Fix the position of $X_1$. There are $31$ remaining seats where $X_3$ can be placed,each with equal probability of $\frac{1}{31}$.$X_1$ and $X_3$ are within earshot if there are at most $3$ persons between them on the minor arc. Let $k$ be the number of persons between $X_1$ and $X_3$ on the minor arc.If $k=0$,$X_3$ is adjacent to $X_1$. There are $2$ such positions (left or right).If $k=1$,there is $1$ person between $X_1$ and $X_3$. There are $2$ such positions.If $k=2$,there are $2$ persons between $X_1$ and $X_3$. There are $2$ such positions.If $k=3$,there are $3$ persons between $X_1$ and $X_3$. There are $2$ such positions.Total favorable positions for $X_3 = 2 + 2 + 2 + 2 = 8$.Total possible positions for $X_3 = 31$.Therefore,the probability is $\frac{8}{31}$.

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