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# Two natural numbers r,s are drawn one at a time,without replacement from the set S=1,2,3,......,n.Find $P[r\leq p|s\leq p],where \;p\in S$

Two number 'r' and 's' are taken successively without replacement from S=$(1\;2\;3\dots\;n$)
Let $E_1$$;$r$\leq$p$;$
$E_2$$;r$$>$p$;$
A$;$event 's' selected$\leq$p
P($E_1/A$)=P($r\leq\;p/s\leq\;p)$
=P(E$_1$/A)$\large\frac{{P(E_1)}{P(A/E_1)}}{{P(E_1)}{P(A/E_1)}+{P(E_2)}{P(A/E_2)}}$
P$E_1$=P($'r'aney number\leq\;p$)
=$\large\frac{p}{n}$
=P($E_2$)=P($r>p$)
=$\large\frac{n-p}{n}$
P($A/E_1$)=P($s\leq\;p/r\leq\;p)$
=$\large\frac{p-1}{n-1}$
P($A/E_2$)=$\large\frac{p}{n-1}$
P($E_1/A$)=$\Large\frac{\frac{p}{n}\times\frac{p-1}{n-1}}{\frac{p}{n}\times\frac{p-1}{n-1}+\frac{n-p}{n}\times\frac{p}{n-1}}$
=$\large\frac{p(p-1)}{p(p-1)+(n-p)p}$
=$\large\frac{(p-1)}{p-1+n-1)}$
=$\large\frac{p-1}{n-1}$

edited Jun 4, 2013