A continent-wide television game show involves $N$ countries. Country $i$ nominates $n_i$ cities including its capital city. Two selection processes are being considered to select the city that will participate first. Process 1 entails selecting a country first and then picking a city of that country. Process 2 entails selecting one city from a pool of all participating cities. Considering that the producers of the show would prefer selecting cities that are not capitals, which process should they choose? You may use the following inequality:
$\sum_{i=1}^Nn_i \sum_{i=1}^N\frac{1}{n_i} \ge N^2$
I don't understand what's the different between Process 1 and 2, can someone help me please?
For the first process if $C$ is the event "a capital city is chosen" we have that
$$ \Pr [C]=\sum_{k=1}^N \Pr [C|X_k]\Pr [X_k]=\frac1{N}\sum_{k=1}^N\frac1{n_k} $$
where $X_k$ represent the event that the $k$-th country is chosen. In the second process we have that
$$ \Pr [C]=\frac{N}{\sum_{k=1}^N n_k} $$
as there are $N$ capital cities in a total of cities of $\sum_{k=1}^N n_k$. Now use the given inequality to conclude.