Hard

Probability

$7$ orbs are labeled $1-7$ and are linked linearly in a vertical stack from the ceiling with orb 1 being a part of the ceiling and orb 7 being closest to the floor. Each orb is attached to adjacent orbs by a chain link. At each time step, one of the remaining links is going to be uniformly at random selected and cut. As a result, all the orbs below that link will fall and shatter. What is the expected number of cuts needed until orb 1 is the only remaining orb?

Notes

Hint