Probability theory

You roll a fair die until the first six appears. What is the expected number of rolls, including the roll that shows the six?

A fair coin is flipped $41$ times. A double is a position $i$ ($1 \le i \le 40$) where flips $i$ and $i + 1$ are both heads; overlapping doubles count separately (HHH contains two). What is the expected number of doubles?

In which of these situations is $E[XY] = E[X]\,E[Y]$ guaranteed? (All variables have finite variance. Choose all that apply.)

Let $S$ be the number of heads in $400$ flips of a fair coin. What is the smallest $c$ for which Chebyshev's inequality guarantees $P(|S - 200| \ge c) \le \tfrac14$?

The proof of the weak law of large numbers has four steps: (1) $E[\bar X_n] = \mu$; (2) $\operatorname{Var} \bar X_n = \sigma^2/n$; (3) apply Chebyshev's inequality to $\bar X_n$; (4) let $n \to \infty$. Which step uses the hypothesis that the $X_i$ are pairwise uncorrelated?

Which statement is a correct consequence of the central limit theorem?

Ten letters are put into ten addressed envelopes uniformly at random. Let $N$ be the number of letters in the right envelope. What is $\operatorname{Var} N$?

Which Lean statement correctly formalises Chebyshev's inequality $c^2\,P(|X - E X| \ge c) \le \operatorname{Var} X$? (Here μ is a probability measure, μ[X] is $E X$ and Var[X; μ] is $\operatorname{Var} X$.)

Write tail(n, k) returning the exact probability, as a Fraction, that the number of heads $S$ in $n$ fair coin flips satisfies $|S - n/2| \ge k$. Then write chebyshev(n, k) returning Chebyshev's bound $\frac{n/4}{k^2}$ as a Fraction. The tests check that the bound always holds and is far from sharp for large $n$.