Randomized BST height analysis : How $Z_{n,i}$ and $Y_{k-1}$ are independent?
I haven't watched the video, but presumably it uses the following model of random BST. You start with one element. Then you insert an element which is smaller than the existing element w.p. 1/2, and larger w.p. 1/2. Then you insert an element which is smaller than both existing elements w.p. 1/3, is between them w.p. 1/3, and larger than both w.p. 1/3. In the $n$th step, you similarly insert the new element in one of the $n$ positions, with equal probability for all.

This means that $Z_{n,k}$, which indicates whether the $n$th element got inserted in the $k$th position, is independent of anything that went on before, and in particular of $Y_1,\ldots,Y_{n-1}$.