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### Introduction

In this post we will be looking at probabilistic coupling, which is putting two random variables (or proceses) on the same probability space. This turns out to be a surprisingly powerful tool in probability and this post will hopefully brainwash you into agreeing with me.

Consider first this baby problem. You and I flip coins and $X_n, Y_n$ denote our $n$-th coin flip. Suppose that I have probability $p_1$ of landing heads and you have probability $p_2 \geq p_1$ of landing heads. Let

$T_Z:=\inf\{n\geq 3: Z_nZ_{n-1}Z_{n-2}=HHH\}$

be the first times $Z$ sees three heads in a row. Now obviously you know that $\mathbb{E}[T_X]\leq \mathbb{E}[T_Y]$, but can you prove it?

Of course here it is possible to compute both of the expectations and show this directly, but this is rather messy and long. Instead this will serve as a baby example of coupling.