Sheldon can't do probability

For the uninitiated, the people mentioned here are characters in the TV programme The Big Bang Theory.

In the episode The Russian Rocket Reaction Howard explains excitedly that some equipment his lab has been working on is going to the International Space Station and that someone needs to go there with it.  "Guess who that someone is?"

Sheldon's reply is "Mohammed Lee" (or other variants of the spelling).

When Howard asks "Who's Mohammed Lee?", Sheldon replies "Mohammed is the most common first name in the world, and Lee the most common surname. As I didn’t know the answer, I thought that gave me a mathematical edge."

This could be a way to introduce pupils to the idea of independent events.

First of all, why does Sheldon think this is a good answer?  If Mohammed is the most common first name in the world and Lee (or Li) is the most common surname then the most common full name is presumably the two of them combined.  Using independent probability, we can multiply p(firstname is Mohammed) by p(surname is Lee).

What is the problem with this reasoning?  The surname Lee/Li is one of the most common names in China and Korea.  The vast majority of people with this surname are likely to have family who originally came from China or Korea.

The boy's name Mohammed (which may be spelled in a number of different ways, including Muhammad and Mohamed), according to Wikipedia has its origins in the Arab World, and is particularly popular in the Middle East, north Africa, Pakistan, Bangladesh and India.

If names were distributed randomly then Sheldon would be right, but given the cultural, religious and geographic ties with names, the two events are not independent and it is likely that there are fewer people who have Mohammed as a forename and Lee as a surname than Sheldon expects.

Thought experiment:  What if we looked at the names of all the pupils who attend a school and found that the most common first name was "Jack" and the most common middle name was "Louise".  How many pupils would be named "Jack Louise"?

I suspect the answer is 'none', although Johnny Cash might know differently.