Question

Two natural numbers r, s are drawn one at a time, without replacement from the set S= 1, 2, 3, ...., n . Find P[r ≤ p|s ≤ p] , where p∈S.

Answer 1

Two number 'r' and 's' are taken successively without replacement from S=(123…n(123…n)Let E1E1;;r≤≤p;;E2E2;r;r>>p;;A;;event 's' selected≤≤pP(E1/AE1/A)=P(r≤p/s≤p)r≤p/s≤p)=P(E11/A)P(E1)P(A/E1)P(E1)P(A/E1)+P(E2)P(A/E2)P(E1)P(A/E1)P(E1)P(A/E1)+P(E2)P(A/E2)PE1E1=P(′r′aneynumber≤p′r′aneynumber≤p)=pnpn=P(E2E2)=P(r>pr>p)=n−pnn−pnP(A/E1A/E1)=P(s≤p/r≤p)s≤p/r≤p)=p−1n−1p−1n−1P(A/E2A/E2)=pn−1pn−1P(E1/AE1/A)=pn×p−1n−1pn×p−1n−1+n−pn×pn−1pn×p−1n−1pn×p−1n−1+n−pn×pn−1=p(p−1)p(p−1)+(n−p)pp(p−1)p(p−1)+(n−p)p=(p−1)p−1+n−1)(p−1)p−1+n−1)=p−1n−1