The OP is not restrictive, 1/9 is for 2 contest wins in a row.

Possible "2 win in a row" posted above are all correct lateral thinking,

but the first thing that comes in mind is that- a contestant winnning

the 1st and 2nd pickings...or 2nd and 3rd pickings is what OP ask.

If the P S R P S R are on satchels no. 1 2 3 4 5 6 respectively

and contestants pick order are c1 c2 c1 c2 c1 c2,

win for c1 has 12 ways (13,16,21,24 . . . 65)for 1st pick;

win2 for c1 has 5 ways (excluding picked satchels) for 2nd pick;

there are 2 ways for pick 3 (ex. 1324w/l,1346t/t,1354w/l,1362w/l,1365w/l)

thus 12x5x2 = 120 for win-win-?,while for ?-win-win it is the same 120 ways.

All ways are equal to permut(6,6)=720 , Chance=240/720 or 1/3

but there is no pick 3 after win-win ...