Chicago • David Griffin rarely plays the lottery, unless one of those enormous jackpots is at stake.

Perhaps he should get in those games more. They seem to like him.

For the fourth time since 2011, the team paying him has won the NBA draft lottery — this time, the New Orleans Pelicans, who bucked seriously long odds Tuesday night and now will have the first chance at selecting Duke forward Zion Williamson next month.

And suddenly, the idea of pairing Anthony Davis with Williamson might just become a reality.

“I don’t believe the universe makes mistakes,” Griffin said. “I just think we have something special going on.”

Griffin was with Cleveland when the Cavaliers won the lottery in 2011, 2013 and 2014. Barely a month into his tenure as New Orleans’ executive vice president of basketball operations, now it’s the Pelicans who have struck lottery gold. They defied the odds to do it; the Pelicans entered with a 6% chance of winning the No. 1 pick for the June 20 draft.

So now, instead of going into a summer where they may have traded Davis — the player New Orleans got when it won the lottery in 2012, the player who was openly disgruntled this season — the Pelicans have a chance at becoming much better, and fast.

“This just jump-starts the process,” Griffin said. “It’ll be harder for me to mess it up than it would have been before this.”

Memphis will choose second, New York third and the Los Angeles Lakers will pick fourth. The Pelicans, Grizzlies and Lakers all moved up to get a top-four spot; New York, Phoenix and Cleveland all entered the night with the best odds of winning the lottery at 14% each, and none of those clubs even got a top-two pick.

The lottery had a new format this year, one that the league hoped would even the playing field in a number of ways and didn’t make all-out tanking a more enticing option to teams looking to maximize their chance at securing the No. 1 pick.

The Knicks, Suns and Cavs didn’t have much interest in winning this season, with none winning more than 19 games. The ping-pong balls apparently weren’t interested in having them win the lottery, either.

“I don’t think you should ever not play to win basketball games,” Pelicans coach Alvin Gentry said. “I don’t see where it would help you.”

Cleveland will pick fifth, followed by Phoenix, Chicago and Atlanta. Washington has the No. 9 pick, Atlanta goes again at 10 and Minnesota is at No. 11. Charlotte is next up, followed by Miami at 13 and Boston at 14. That pick was conveyed to the Celtics by Sacramento as part of an earlier trade.

The Hawks felt they were big winners, too, after getting two top-10 picks.

“There are always good players,” said Hawks guard Kevin Huerter, the No. 19 pick in last year’s draft and part of Atlanta’s contingent at the lottery. “Every team just has to go out and find them.”

UCF center Tacko Fall, the 7-foot-6 draft hopeful, played against Williamson in the second round of the NCAA tournament — a 77-76 Duke victory that went down to the very last moment, a game where Williamson scored 32 points. Fall was asked Tuesday who he would take with the No. 1 pick in this draft, and he did not hesitate for even a second before answering.

“Zion,” Fall said. “He’s a once-in-a-generation player. I’d seen him on TV a lot, but when you play against this kid, you just see it. He’s different.”

Murray State’s Ja Morant, projected by many as a strong candidate to be chosen No. 2 overall, was among the players in the audience viewing the proceedings.

“I don’t have any reaction,” Morant said. “I was just excited to be here and be in the position that I’m in and just to see how it all played out. Honestly, I don’t know what number or where I’ll land. Obviously, whatever team drafts me, I’ll be happy to go there and bring it every night.”

Williamson was one of three now-former Duke starters at the lottery — R.J. Barrett and Cam Reddish were among the invited players as well.

Barrett and Reddish don’t know where they’re heading.

Williamson now has a pretty good idea.

Aided by a lucky tie that came from Cavaliers executive Jeff Cohen — the one that Cohen wore for all three of Cleveland’s lottery wins, and one that Griffin insisted he get to use this year — along with an angel pin that belonged to his grandmother and the winning ping-pong ball combination of 7-4-12-13, Griffin is already off to a flying start in New Orleans.

And maybe Davis will be more interested in staying now.

“I don’t think it’ll play into it hugely,” Griffin said. “If he was open-minded to believing that we could build a winner around him, he’s more open-minded to it. ... We’re going to build something that we hope everyone wants to be part of and I believed very strongly Anthony’s going to want to be part of that whether we win this or not. I think when you have elite talent, that tends to attract other truly elite talent.”

NBA draft order
Thursday, June 20
At Barclays Center, Brooklyn, N.Y.
First Round
1. New Orleans
2. Memphis
3. New York
4. L.A. Lakers
5. Cleveland
6. Phoenix
7. Chicago
8. Atlanta
9. Washington
10. Atlanta (from Dallas)
11. Minnesota
12. Charlotte
13. Miami
14. Boston (from Sacramento through Philadelphia)
15. Detroit
16. Orlando
17. Brooklyn
18. Indiana
19. San Antonio
20. Boston (from L.A. Clippers through Memphis)
21. Oklahoma City
22. Boston
23. Utah
24. Philadelphia
25. Portland
26. Cleveland (from Houston)
27. Brooklyn (from Denver)
28. Golden State
29. San Antonio (from Toronto)
30. Milwaukee
Second Round
31. Brooklyn (from New York through Philadelphia)
32. Phoenix
33. Philadelphia (from Cleveland through Orlando and New York)
34. Philadelphia (from Chicago through L.A. Lakers)
35. Atlanta
36. Charlotte (from Washington through Orlando through Denver through Atlanta)
37. Dallas
38. Chicago (from Memphis)
39. New Orleans
40. Sacramento (from Minnesota through Portland and Cleveland)
41. Atlanta (from L.A. Lakers through Cleveland and Indiana)
42. Philadelphia (from Sacramento through Brooklyn and Milwaukee)
43. Minnesota (from Miami through Charlotte)
44. Atlanta (from Charlotte)
45. Detroit
46. Orlando (from Brooklyn through Memphis and Charlotte)
47. Sacramento (from Orlando through New York)
48. L.A. Clippers
49. San Antonio
50. Indiana
51. Boston
52. Charlotte (from Oklahoma City)
53. Utah
54. Philadelphia
55. New York (from Houston)
56. L.A. Clippers (from Portland through Detroit and Orlando)
57. New Orleans (from Denver through Milwaukee)
58. Golden State
59. Toronto
60. Sacramento (from Milwaukee)