Question: 2022 Winter Final - 15

Author: Michiel Smid

Let $\pi_1,\ldots,\pi_{20}$ be a random permutation of $\{1,\ldots,20\}$. Define the events: $ A = \pi_{10}>\pi_{11} $ and $ B = \pi_{11} \gt \pi_{12} $ Which of the following is true?

(a)

$\Pr(A\mid B)$ is undefined

(b)

$\Pr(A\cap B) < \Pr(A)\cdot\Pr(B)$

(c)

None of the above

(d)

$\Pr(A\cap B) = \Pr(A)\cdot\Pr(B)$

(e)

$\Pr(A\cap B) > \Pr(A)\cdot\Pr(B)$