Hiring problem from CLRS
In fact, your result is correct and that reference solution is wrong!

Here is where that reference solution goes wrong.


  Since we view the candidate ranking as reading an random permutation, this is equivalent to the probability that a random permutation is a decreasing sequence followed by an increase, followed by another decreasing sequence.


A permutation that leads to hiring of two candidates if and only if the first candidate is not the best candidate and is better than all others candidates that come before the best candidate. There is no requirement of sequences being decreasing or increasing.

(There is a minor typo in your case 2. You meant 3/4 instead of 1/4.)

I would not say your reasoning is fully correct yet, though you have got the correct result, since I have not found a natural way to explain rigorously your way of computing the probability of each case by a product of probabilities. I am even more far away from being certain that your reasoning is wrong. It looks like you shared my feeling as well.

There are better or more rigorous ways to produce the correct result. I will let you think about it.