How many people have their first and last name beginning with the same letter?

Question

In a country, with a population of 11.000.000 people, how many of them have their first and last name beginning with the same letter? (e.g. Alex Abus, Peter Pen etc.) . Consider that their alphabet has 26 letters and that everyone has exactly one first name and one last name. I have tried the upper problem and the following is my "solution". Can somebody please ensure me that is right or if its wrong explain to me the reason and give me the right solution?

Considering the Generalized Pigeonhole Principle:
at least
11.000.000 / 26 = 423.077 people have their first name beginning with the same letter.

From these 432.077 people, 
at least
432.077 / 26 = 16.619 people have their first AND last name beginning with the same letter.

Answer 1

If you assume that the beginning letters of first and last names are uniformly chosen from $26$ letters and chosen independently, $\frac 1{26}$ of the people will have matching first and last initials. If the initial letters are chosen independently from some distribution the chance will be at least this (Imagine that $90\%$ of all names start with A and the rest start with other letters. $81\%$ of people would then have initials AA). Without more information about the distribution I don't think you can say more than this. Maybe every first name starts with A and every last name starts with B. Then nobody has matching initials.

Your calculation comes from the assumptions in my first sentences, but results in the number with both letters the same and a given letter of the alphabet. I read the problem to ask how many matches there are, regardless of which letter a person uses.

Answer 2

Considering the Generalized Pigeonhole Principle: at least 11.000.000 / 26 = 423.077 people have their first name beginning with the same letter.

Okay, Let's say that letter is $M$. Let's say there are 526,217 people whose first name starts with $M$.

From these 432.077 people, at least 432.077 / 26 = 16.619 people have their first AND last name beginning with the same letter.

But that letter does not have to be $M$.

That means among the 526,217 people whose first name starts with $M$, the most common letter for the last name (let's suppose it is $J$) will have at least $16,619$ people having. It so let's suppose there are $27,145$ people with the initial M.J.

But it's perfectly conceivable that of the $526,217$ people whose first name begins with $M$ that NONE of them have last names beginning with $M$.

......

Thought experiment. Let's say the King of this Country had a traumatic experience with comic books as child and decreed "As I hate Louis Lane, Lana Lang, and Peter Parker, it will be illegal in this country for anyone to have matching initials. Any family which names their child with a first name that begins with the same letter as their last name shall be put to death."

In that case the answer is $0$.

Or if the King said "It will be required by law, that every child must have a first name with the same initial as the last name" then the answer would be $11,000,000$.

======== old answer =========

To answer "At least how many people share a pair of initials":

There are $26^2$ possible pairs of initials. It's not possible for every pair of initials to have fewer than $\frac {11\times 10^6}{26^2}$ people with it, so at least one pair will have at least $\lceil \frac{10^6}{26^2} \rceil$ people with that pair.

No need to do it twice for each name.

Thus the most popular pair of initials will have at least that many.

...

BUT THAT WASN'T THE QUESTION

how many of them have their first and last name beginning with the same letter? (e.g. Alex Abus, Peter Pen etc.)

This is impossible to answer without knowing how the names are distributed.

As the initials $A$ and $M$ are much more common than $X$ or $Z$ and as the initials $A$ and $E$ are more common for first names than they are for last names, and as there is simply no-way any one of us can actually know the distribution, and as the distribution is so obviously NOT normally distributed, I simply can't see this as a valid question.

If we lived on another planet an we were TOLD initials are normally distributed given and JFOWGKEL JFKOW is just as common a name as ANTHONY SMITH then we'd expect one out of 26 people or $\frac {11,000,000}{26}$ people to have alliterative initials.