N.Y. Comp. Codes R. & Regs. tit. 9, § 6204.3 – Methods for determining ballot order by lot.

(a) Whenever a Board of Elections, or other elections officer prepares a primary ballot, or receives a written request from a candidate to determine ballot order of candidate names on a ballot, or the order of names and emblems of independent bodies on a ballot, or prepares a ballot for a primary election, they shall use one of the following methods:
1. (1) acquire a quantity of smooth-surfaced, numbered spheres and an automatic selection machine similar to those used for bingo games or for drawing state lottery numbers. Place all spheres in the automatic machine. The person(s) designated to conduct the drawing shall release one sphere. The number drawn will be noted on the candidate list, next to the name of the candidate. Once the drawing for that office is complete, the numbers on the candidate list will be reviewed, and the ballot order announced, with the lowest number representing the first position, the next lowest number representing the number two position, and so on;
2. (2) acquire a quantity of smooth-surfaced, numbered spheres, of the type available from any bingo supplier or as available for use by the public in helping to select lottery numbers. Place all spheres in a container with an opening narrow enough to prohibit the insertion of a hand, and wide enough to allow the release of no more than one sphere at a time. The person(s) designated to conduct the drawing shall release one sphere. The number drawn will be noted on the candidate list, next to the name of the candidate. Once the drawing for that office is complete, the numbers on the candidate list will be reviewed, and the ballot order announced, with the lowest number representing the first position, the next lowest number representing the number two position, and so on; or
3. (3) acquire a quantity of smooth-surfaced, numbered spheres, of the type available from any bingo supplier or as available for use by the public in helping to select lottery numbers. Place as many spheres, sequentially numbered, as there are candidates or independent bodies in the particular drawing, in a container with an opening narrow enough to prohibit the insertion of a hand, and wide enough to allow the release of no more than one sphere at a time. The person(s) designated to conduct the drawing shall release one sphere. The number drawn shall designate that candidate's position on the ballot, and will be so noted on a candidate list, next to the name of that candidate.
(b) Prior to the date of the drawing, the commissioners shall designate at least two persons who will conduct the drawing for all offices. Candidates or their designees may inspect the device or devices to be used for the drawings, at the date and time established by the board, and in the presence of the two commissioners or their designees.