The odds of a singleton card appearing the draft pool are 360/540 = 2/3 = ~67%. The odds of a player opening the card directly in one of their packs is 45/540 = ~8.3% (or 45/360 = 12.5% if you know the card is in the draft pool and not in the leftovers pile). The odds of a specific player seeing the card are slightly more difficult to calculate because you have to account for whether the card gets drafted by another player or passed to you.
Total Pool: 540
8 player draft > 3 x 15
Distribution is 360 total
210 left over...
probability of "x = 1" being drawn in the 360?
...stumped.