Can You Cut The Perfect Pancake?

Riddler Classic

In a huge electorate, every voter independently picks candidate $A$ or $B$ with probability $\tfrac12$. On election night the $80\%$ who voted in person are counted; the remaining $20\%$ (mail) are counted later. What is the probability that whoever trailed on election night ends up winning?

Solution

Measure each candidate’s lead as a margin (votes for $A$ minus votes for $B$). With a vast electorate the night margin $D_n$ (from $80\%$ of voters) and the mail margin $D_m$ (from $20\%$) are independent, each a zero-mean normal with variance proportional to its share: $\operatorname{Var}(D_n) = 0.8$, $\operatorname{Var}(D_m) = 0.2$ in units of the total vote count. The final margin is $D_f = D_n + D_m$, with variance $1$.

The night-trailer wins exactly when $D_n$ and $D_f$ have opposite signs. The pair $(D_n, D_f)$ is jointly normal with correlation $$\rho = \frac{\operatorname{Cov}(D_n, D_f)}{\sqrt{\operatorname{Var}(D_n)\operatorname{Var}(D_f)}} = \frac{0.8}{\sqrt{0.8 \cdot 1}} = \sqrt{0.8}.$$ For a zero-mean bivariate normal the chance of opposite signs is $\tfrac12 - \tfrac{1}{\pi}\arcsin\rho$, so $$\Pr(\text{trailer wins}) = \frac12 - \frac{\arcsin\sqrt{0.8}}{\pi} = \frac12 - \frac{\arctan 2}{\pi} \approx \boxed{0.1476}.$$ A late lead change happens about one race in seven: the mail fifth of the vote can overturn the night result, but only when election night was already close.

The computation

Simulate a large electorate: split the vote into the in-person $80\%$ and mail $20\%$, draw each as a fair binomial, and count how often the night margin and the full margin disagree in sign.

import numpy as np
rng = np.random.default_rng(0)
N, runs = 10_000_000, 2_000_000
nd, nm = int(0.8 * N), int(0.2 * N)
day = rng.binomial(nd, 0.5, runs) - nd / 2          # A's election-night margin
mail = rng.binomial(nm, 0.5, runs) - nm / 2         # A's mail margin
flip = np.mean(np.sign(day) != np.sign(day + mail))
print(flip, 0.5 - np.arctan(2) / np.pi)             # ~0.1476