Probabilistic techniques, methods, and ideas in ("undergraduate") real analysis As the book Probabilistic Techniques in Analysis by Richard F. Bass shows, nowadays techniques drawn from probability are used to tackle problems in analysis. 
The mentioned book presents a survey of these methods "at the level of a beginning Ph.D. student", but I would like to see some examples of "more basic" applications of probability to undergraduate (so to say) real analysis (in other words, I would like to see some applications of probabilistic reasoning to calculus problems).

Here is an example.
The question is how to show that
$$\binom{n}{k}^{-1}=(n+1)\int_0^1 x^k (1-x)^{n-k} \, dx. $$
To make this self-contained, I'll paste this answer below:
Let's do it somewhat like the way the Rev. Thomas Bayes did it in the 18th century (but I'll phrase it in modern probabilistic terminology).
Suppose $n+1$ independent random variables $X_0,X_1,\ldots,X_n$ are uniformly distributed on the interval $[0,1]$.
Suppose for $i=1,\ldots,n$ (starting with $1$, not with $0$) we have:
$$Y_i = \begin{cases} 1 & \text{if }X_i<X_0 \\ 0 & \text{if }X_i>X_0\end{cases}$$
Then $Y_1,\ldots,Y_n$ are conditionally independent given $X_0$, and $\Pr(Y_i=1\mid X_0)= X_0$.
So $\Pr(Y_1+\cdots+Y_n=k\mid X_0) = \dbinom{n}{k} X_0^k (1-X_0)^{n-k},$ and hence
$$\Pr(Y_1+\cdots+Y_n=k) = \mathbb{E}\left(\dbinom{n}{k} X_0^k (1-X_0)^{n-k}\right).$$
This is equal to
$$
\int_0^1 \binom nk x^k(1-x)^{n-k}\;dx.
$$
But the event is the same as saying that the index $i$ for which $X_i$ is in the $(k+1)$th position when $X_0,X_1,\ldots,X_n$ are sorted into increasing order is $0$.
Since all $n+1$ indices are equally likely to be in that position, this probability is $1/(n+1)$.
Thus $$\int_0^1\binom nk x^k(1-x)^{n-k}\;dx = \frac{1}{n+1}.$$
