Exercise 5.2.2

In HIRE-ASSISTANT, assuming that the candidates are presented in a random order, what is the probability that you hire exactly twice?

You hire twice when you first hire is the candidate with rank i and all the candidates with rank k > i come after the candidate with rank n.There are n-i better suited candidates and the probability of the best one coming first is 1/(n-i) (we can ignore the other candidates and they don't affect the probability). Thus, the probability for hiring twice if your first candidate has rank i is:
$$\Pr{T_i} = \frac{1}{n}\frac{1}{n-i}$$
The first part reflects the probability of picking that particular candidate out of n.
The probability to hire twice is:
$$\Pr{T} = \sum_{i=1}^{n-1}\Pr{T_i}
= \sum_{i=1}^{n-1}\frac{1}{n}\frac{1}{n-i}
= \frac{1}{n} \sum_{i=1}^{n-1}\frac{1}{i}
= \frac{1}{n} \Big(\lg(n-1) + O(1)\Big)$$