Probability probability
"Two Children Puzzle / Boy Born on a Tuesday" and variants:
A "liar paradox" variant: Multiple-choice question about the probability of a random answer to itself being correct
A bag that contains a coin with two heads, a coin with two tails, and a standard coin; one is brought out and flipped; it shows heads. What is the probability that it is the standard coin?
Expectation as tail probability (complementary CDF) integral/sum.
Continuous (generalized to any $E[X^p]$ in the answer): Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$.
Discrete: Find the Mean for Non-Negative Integer-Valued Random Variable
Measure-theoretic treatment: Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$
Statistics statistics
Upper tail inequality for the standard normal distribution:
Normal approximation (CLT) for discrete Binomial: ... (to be continued... okay, this task is more complicated than expected)