On the Expected Duration of a Generalized Bingo Game

TL;DR

The paper generalizes Bingo to an (n,m)-Bingo with an n x n card and m possible values per column, and derives exact expressions for the distribution and expected duration using the inclusion-exclusion principle. For a single card, the authors prove that the expected number of calls satisfies E[B] = (mn+1)(1 - S_n) where S_n = ∑_{X ≠ ∅} (-1)^{|X|+1} 1/(m(X)+1), which implies E[B] is linear in m with slope n(1 - S_n). The multiplayer extension conditions on the realized set of unique lines and yields E[B_N | L_N] = (mn+1)(1 - S_{n,N}) with S_{n,N} defined analogously, again exhibiting linearity in m and enabling numerical validation that aligns closely with Monte Carlo simulations. The results provide exact frameworks for durations of Bingo variants and open avenues for asymptotic analysis, variance, and fast approximations, with potential applicability to other line-based winning rules.

Abstract

We investigate the expected number of calls required to achieve Bingo in a generalized (n,m)-Bingo game, where each n x n card is filled by sampling n numbers from m possible values per column. Using the inclusion-exclusion principle, we derive exact formulas for the probability distribution and the expected game length. Our main theoretical result proves that the expected number of calls is a linear function of m.

Figures (2)

• Figure 1: A $5 \times 5$ Bingo Card
• Figure 2: Expected number of calls $\mathbb{E}[\mathbf{B}]$ versus $m$ for $n=5$ and $n=7$.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof