In Time Travel and Other Mathematical Bewilderments, Martin Gardner presents a set of four nontransitive bingo cards designed by Donald Knuth (pp. 61). The rules are that the first player to complete a horizontal row wins. Gardner does not delve into the mathematics but merely mentions that, probabilistically, A beats B, B beats C, C beats D, and D beats A.
I was so baffled that I immediately went on to to verify those results. He was right.
Now, with dice or heads tails sequences, I can understand nontransitivity; can someone please give the mathematical explanation of how and why nontransitivity occurs in the above game.
Let's say two players take cards $A$ and $B$ and we roll a 6-sided die to call out the numbers.
If a $1$ or a $3$ is rolled right off the start, then player $B$ cannot win using their top row and must rely on a $5,6$ victory. Player $A$ does not experience this problem if something on player $B$'s card is rolled first.