Lets get it by contradiction.
The negation of the
Either P1 is there and not P2 in the 8 winners or vice-versa
is
Either both P1 and P2 are there in 8 winners or none of P1 and P2 is there is 8 winers
.................................(1)
Consider the probablity for the negation.
Events where both P1 and P2 are not there in 8 winners = $\binom{14}{8}$
Events where both P1 and P2 are there in the 8 winners = $\binom{14}{6}$
#ways to select the 8 winners = $\binom{16}{8}$
Now, lets evaluate the probablity for the statement (1).
P( statement (1) ) = ( $\binom{14}{8}$ + $\binom{14}{6}$ ) / ( $\binom{16}{8}$ )..............(2)
P(Either P1 is there and not P2 or vice-versa) = 1 - answer in statement (2)
= 8/15