"Yessir, ladies and gentlemen, he's one of the chosen people!" (26)
This line from the bingo caller helps reinforce the notion that the bingo game is deterministic. It controls people's fate. At the same time, however, the line carries a subtle undertone – the protagonist is, in fact, chosen and marked by his race.
He felt vaguely that his whole life was determined by the bingo wheel; not only that which would happen now that he was at last before it, but all that had gone before, since his birth and his mother's birth and the birth of his father. (32)
The protagonist suddenly blames the bingo wheel for all the oppression he has experienced in his life (and that his ancestors have experienced). This fits in with the bingo wheel as the wheel of fortune (see "Symbols, Imagery, Allegory").
I better get down from here before I make a fool of myself, he thought. (32)
Does this represent a moment where the protagonist might have exercised his free will?
He steeled himself; the fear had left, and he felt a profound sense of promise, as though he were about to be repaid for all the things he'd suffered all his life. Trembling, he pressed the button. (42)
This moment represents the protagonist's attempted mastery over his fortune and fate.
"Didn't they know that although he controlled the wheel, it also controlled him, and unless he pressed the button forever and forever and ever it would stop, leaving him high and dry, dry and high on this hard high slippery hill and Laura dead?" (65)
If the wheel is a stand-in for fate (see "Symbols, Imagery, Allegory"), what might it mean that the wheel (fate) controls him? We might think about how a man is subject to his own fate, like it or not, and how the protagonist's control of the wheel suggests an element of free will.