Suppose that you are arranging a chain of n dominos so that, once you are done, you can have them...

Suppose that you are arranging a chain of n dominos so that,
once you are done, you can have them all fall sequentially in a pleasing manner
by knocking down the lead domino. Each time you try to place a domino in the
chain, there is some chance that it falls, taking down all of the other dominos
you have already carefully placed. In that case, you must start all over again
from the very first domino.

(a) Let us call each time you try to place a domino a trial.
Each trial succeeds with probability p. Using Wald’s equation, find the
expected number of trials necessary before your arrangement is ready. Calculate
this number of trials for n = 100 and p = 0.1.

(b) Suppose instead that you can break your arrangement into
k components, each of size n/k, in such a way so that once a component is
complete, it will not fall when you place further dominos. For example: if you have
10 components of size 10, then once the first component of 10 dominos are
placed successfully they will not fall; misplacing a domino later might take
down another component, but the first will remain ready. Find the expected
number of trials necessary before your arrangement is ready in this case.
Calculate the number of trials for n = 100, k = 10, and p = 0.1, and compare
with your answer from part (a).