The probabilistic method in combinatorics As I understand it, a typical pattern of argument is as follows. We have a set $X$ and want to show that at least one element of $X$ has property $P$. We choose some function $f: X \to \mathbb{N}$ such that $f(x) = 0$ iff $x$ satisfies $P$, and we choose a probability measure on $X$. Then we show that with respect to that measure, $\mathbb{E}(f) < 1$. It follows that $f^{-1}\{0\}$ has positive measure, and is therefore nonempty.

Real analysis One example is Banach's proof that any measurable function $f: \mathbb{R} \to \mathbb{R}$ satisfying Cauchy's functional equation $f(x + y) = f(x) + f(y)$ is linear. Sketch: it's enough to show that $f$ is continuous at $0$, since then it follows from additivity that $f$ is continuous everywhere, which makes it easy. To show continuity at $0$, let $\varepsilon > 0$. An argument using Lusin's theorem shows that for all sufficiently small $x$, the set $\{y: |f(x + y) - f(y)| < \varepsilon\}$ has positive Lebesgue measure. In particular, it's nonempty, and additivity then gives $|f(x)| < \varepsilon$.

Another example is the existence of real numbers that are normal (i.e. normal to every base). It was shown that almost all real numbers have this property well before any specific number was shown to be normal.

Set theory Here I take ultrafilters to be the notion of measure, an ultrafilter on a set $X$ being a finitely additive $\{0, 1\}$-valued probability measure defined on the full $\sigma$-algebra $P(X)$. Some existence proofs work by proving that the subset of elements with the desired property has measure $1$ in the ultrafilter, and is therefore nonempty.

One example is a proof that for every measurable cardinal $\kappa$, there exists some inaccessible cardinal strictly smaller than it. Sketch: take a $\kappa$-complete ultrafilter on $\kappa$. Make an inspired choice of function $\kappa \to \{\text{cardinals } < \kappa \}$. Push the ultrafilter forwards along this function to give an ultrafilter on $\{\text{cardinals } < \kappa\}$. Then prove that the set of inaccessible cardinals $< \kappa$ belongs to that ultrafilter ("has measure $1$") and conclude that, in particular, it's nonempty.

(Although it has a similar flavour, I would not include in this list the cardinal arithmetic proof of the existence of transcendental real numbers, for two reasons. First, there's no measure in sight. Second -- contrary to popular belief -- this argument leads to an explicit construction of a transcendental number, whereas the other arguments on this list do not explicitly construct a thing with the desired properties.)

The probabilistic method in combinatorics As I understand it, a typical pattern of argument is as follows. We have a set $X$ and want to show that at least one element of $X$ has property $P$. We choose some function $f: X \to \{0, 1, \ldots\}$ such that $f(x) = 0$ iff $x$ satisfies $P$, and we choose a probability measure on $X$. Then we show that with respect to that measure, $\mathbb{E}(f) < 1$. It follows that $f^{-1}\{0\}$ has positive measure, and is therefore nonempty.

Real analysis One example is Banach's proof that any measurable function $f: \mathbb{R} \to \mathbb{R}$ satisfying Cauchy's functional equation $f(x + y) = f(x) + f(y)$ is linear. Sketch: it's enough to show that $f$ is continuous at $0$, since then it follows from additivity that $f$ is continuous everywhere, which makes it easy. To show continuity at $0$, let $\varepsilon > 0$. An argument using Lusin's theorem shows that for all sufficiently small $x$, the set $\{y: |f(x + y) - f(y)| < \varepsilon\}$ has positive Lebesgue measure. In particular, it's nonempty, and additivity then gives $|f(x)| < \varepsilon$.

Another example is the existence of real numbers that are normal (i.e. normal to every base). It was shown that almost all real numbers have this property well before any specific number was shown to be normal.

Set theory Here I take ultrafilters to be the notion of measure, an ultrafilter on a set $X$ being a finitely additive $\{0, 1\}$-valued probability measure defined on the full $\sigma$-algebra $P(X)$. Some existence proofs work by proving that the subset of elements with the desired property has measure $1$ in the ultrafilter, and is therefore nonempty.

One example is a proof that for every measurable cardinal $\kappa$, there exists some inaccessible cardinal strictly smaller than it. Sketch: take a $\kappa$-complete ultrafilter on $\kappa$. Make an inspired choice of function $\kappa \to \{\text{cardinals } < \kappa \}$. Push the ultrafilter forwards along this function to give an ultrafilter on $\{\text{cardinals } < \kappa\}$. Then prove that the set of inaccessible cardinals $< \kappa$ belongs to that ultrafilter ("has measure $1$") and conclude that, in particular, it's nonempty.

(Although it has a similar flavour, I would not include in this list the cardinal arithmetic proof of the existence of transcendental real numbers, for two reasons. First, there's no measure in sight. Second -- contrary to popular belief -- this argument leads to an explicit construction of a transcendental number, whereas the other arguments on this list do not explicitly construct a thing with the desired properties.)