Bingo balls are like Lotto balls. They're like ping-pong balls with numbers on them. A ball gets picked somehow and that becomes the winning number, or the number that is called out in Bingo.
Before the winning ball gets picked, all the eligible balls are in some kind of container which can be rotated to make sure the balls are mixed. When a ball gets picked we say it is "drawn," in the manner of drawing a card from a shuffled deck of cards. Hence "draw-tank."
I expect the allusion is comparing the future of a child to the game of chance. "His cards have not yet been drawn."