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A coupling of probability distributions $\mu ,\nu$ is a joint probability distribution of random variables $(X,Y)$ whose marginal distributions are $\mu ,\nu$. It lets us choose the statistical dependence between two random constructions while preserving the individual laws. This choice can make a comparison much easier: giving two simulations the same random input may keep their paths close, even though separately generating both paths would hide that relationship.^[ref]

For example, two fair coin tosses can be generated independently, with mismatch probability $1/2$, or by tossing once and setting $Y=X$, with mismatch probability zero. Both constructions have the same two Bernoulli distributions as marginals. Their joint laws differ. A maximal coupling makes the values agree as often as possible; a coalescing coupling arranges permanent agreement of two processes after a random time. Minimizing expected geometric separation instead leads to Wasserstein distance.

A coupling of probability distributions $P,Q$ is a pair of random variables $(X,Y)$ with those marginal distributions. On a finite set, and more generally on a standard Borel space, the least achievable mismatch isA maximal coupling attains the minimum; maximal here refers to agreement. An arbitrary coupling can have a much larger mismatch.^[ref]

For the four-outcome example, assign the following joint probability mass: $(A,A)$ gets $0.1$, $(B,B)$ gets $0.1$, $(C,C)$ gets $0.2$, and $(D,D)$ gets $0.3$. The remaining $0.3$ must have $X=A$; place $0.2$ on $(A,B)$ and $0.1$ on $(A,C)$. Row sums give $P$, column sums give $Q$, and the diagonal contains $0.7$. Agreement can reach 70%; mismatch cannot fall below 30%.

To see the lower bound, for any event $E$,The two indicator functions agree whenever $X=Y$. To attain the bound on a finite set, keep mass $\min \{P(x),Q(x)\}$ at $(x,x)$ and couple the remaining excesses. The two residual probability distributions live on disjoint subsets, so their part necessarily mismatches. When the probability distributions are identical, the residual part disappears entirely.

This is a choice of joint construction, not a claim that independently drawn observations are usually equal. Two independent random variables with the same continuous probability distribution can disagree with probability one even though their total variation distance is zero. The construction of a maximal coupling deliberately introduces statistical dependence to reveal how similar the marginal laws are.

Total variation distance asks how much two probability distributions can disagree about the same event. For probability measures $P,Q$ on one measurable space $(S,\mathcal{F})$, its definition isThe supremum ranges over all measurable sets, so $d_{\mathrm{TV}}(P,Q)\le \epsilon$ guarantees that every yes-or-no prediction has probability error at most $\epsilon$. This metric takes values between zero and one; it is zero exactly when the two probability measures agree. This makes it useful for comparing a sampling algorithm's output with its target probability distribution, and for defining the mixing time of a Markov chain.

On a finite set, add the excess probability mass wherever $P$ exceeds $Q$. That excess is also half the sum of all absolute differences:The last expression reveals the common probability mass that the two probability distributions can share.^[ref] The coupling characterization of total variation distance turns this overlap into an actual joint construction; minimum sum of errors in a simple hypothesis test gives it a statistical interpretation. One quantity connects probability bookkeeping, comparison of sampling methods, and the limits of a statistical test.