That verse rarely appears in tradition, but there's a form of it in Child 24A, Bonnie Annie, verse 7:

They've casten black bullets twice six and forty,
And ae the black bullet fell on bonnie Annie.

That text was taken from Kinloch's Ancient Scottish Ballads. The sequence also appears in a set noted by Sabine Baring-Gould in Devon, The Undutiful Daughter. This is in his MSS in collated form, the tune and the bulk of the text being from J. Masters, Bradstone, 1888; with additional material (including, it seems, this verse, no. 5 in the MS) from H. Smith, Two Bridges, 1889.

They cast the black bullets, as they sailed on the water
The black bullet fell to the undutiful daughter.
Now, who in the ship must go over the side O
O none save the maiden, the fair captain's bride O.

The full song is in Bronson, I, no. 24.7, pp. 300-301. It can also be seen in pdf format at Martin Graebe's Sabine Baring-Gould and the folk songs of South-West England:

The Undutiful Daughter.

Martin Carthy recorded a collated form of the song that features the incident. In his case, the tune and the bulk of the text were from David Clements of Basingstoke, Hampshire (recorded by Ralph Vaughan Williams, January 1909), with three verses adapted from Kinloch. Clements' set was printed in The Journal of the Folk Song Society, III (4) 1909 292-3 and Bronson I 25.15 303. A cylinder recording presumed to be of him survives at the Vaughan Williams Memorial Library, and a transcription can be heard on the EFDSS CD A Century of Song (EFDSSCD02, 1998).

The process is one of drawing lots; evidently in this case using bullets; one only being black.