How to earn Play Coins

The tickets are not random, so your latter assumption is correct. The numbers are predetermined and will give you a certain prize based on which number you had. There are a few numbers (4, 6, 16, 30, 38) which will only give you a random piece of regular furniture, carpet or wallpaper rather than one of the unique items.

The full list is as follows:

1. Mushroom Mural
2. Block Floor
3. Hero's Tunic
4. Random "regular" furniture, wallpaper, or carpet
5. Hero's Pants
6. Random "regular" furniture, wallpaper, or carpet
7. Peach's Parasol
8. Hero's Cap
9. Majora's Mask
10. Samus Mask
11. Midna's Mask
12. Toad Hat
13. Red Pikmin Hat
14. Blue Pikmin Hat
15. Yellow Pikmin Hat
16. Random "regular" furniture, wallpaper, or carpet
17. Brick Block
18. Coin
19. Flagpole
20. Fire Flower
21. Super Mushroom
22. Koopa Shell
23. Starman
24. 1-Up Mushroom
25. ? Block
26. Green Pipe
27. Fire Bar
28. Cannon
29. Yoshi's Egg
30. Random "regular" furniture, wallpaper, or carpet
31. Triple Shells
32. Kart
33. (Pikmin)
34. Blue Falcon
35. Master Sword
36. Arwing
37. Triforce
38. Random "regular" furniture, wallpaper, or carpet
39. Dolphin
40. Virtual Boy
41. Wii Balance Board
42. Varia Suit Pants
43. Varia Suit Shoes
44. Bad Bro 'Stache
45. Big Bro Mustache
46. Varia Suit Body
47. Metroid
48. Triple Bananas
49. Hero's Boots
50. Fay's Mask 

-Source

There is indeed a certain order in which you receive the items, up to a point. Afterwards, you receive one of the previously awarded items randomly. The order is as follows:

First time: Nothing
Second time:  Balloon
Third time:   Balloon
Fourth time:  Pinwheel
Fifth time:   Bunny Balloon
Sixth time:   Bubble Wand
Seventh time: Heart Balloon
Eighth time:  Ice Cream
Ninth time:   Soft-serve Ice Cream
Tenth time:   Party Blower
More than ten times: Random item listed above

The color of the items you receive is random. You also can only Streetpass the same person every 8 hours to receive these gifts.

Additional Sources