In my family, a situation has arisen during the last few years.  This
weekend, a few of us were trying to determine the chances of one
aspect of this situation happening.

TARGET RESEARCHER: Mathematician or statistitian

SITUATION: I have two younger sisters, Jessica and Christy (not their
actual names).  I recently found out that I have an older half sister
ALSO named Christy (who we can refer to as Christy2). She was adopted
by a foster family and named by them. There is a 12-year difference
between the ages of Christy and Christy2.  We are all born in the
continential USA.  Both of their names are spelled the same.  I've
seen their names spelled three different ways, though the spelling
they have is by far the most common.

QUESTION: What are the chances that my younger sister, Christy, and my
half sister, Christy2, were given the same name?  The answer should
include the factors and numbers used in the calculation (i.e. number
of baby names, number of children in our family, etc).  It is not
necessary to include factors regarding ethnicity.  According to
http://www.ssa.gov/OACT/babynames/ with regard to the name popularity
for the year they were each born, Christy was ranked 15 and Christy2
at 4.

This is not as unlikely as it may seem. To get a precise answer for
the probability, we have to formulate a precise question. Here is one
such question:

Q. Assuming that girl's names are selected randomly with a
distribution given by the actual percentages, what is the probability
that the younger sister was given the same name as her older half
sister?

Since the younger sister had the 15th most common name for that year,
we need to now the percentage of girls given that name. Arbitrarily
picking 1995, the 15th most popular name, Lauren, was given to 0.7% of
the girls born that year, so the probability of your younger sister
getting that name is 0.7% or about one in 140, not a very remarkable
occurrence.

We could formulate the question slightly differently, but the answers
will be similar. For example if I had asked the same question, but
swapped the two sisters, I would have gotten a slightly larger
probability, because the 4th most popular name is a bit over 1% of the
population. It is more complex to compute an answer for a symmetrical
version of the question, but the answer will be between the two,
somewhere between 1 in 140 and 1 in 90.

kottekoe aswered the question from the standpoint that your sisters'
names are given.  However, your question asks about the chances of two
people having the same name.  I assume you would have been equally
curious if the older sister and the half sister shared the same name. 
Or if the name shared was "Mary" rather than "Christy".  Taking all of
this into account would involve a considerable amount of calculation. 
You would have to run thru all the female names and the chance for
each to have them.  Then you would have to approximately double it
because the match could have been with either sister.

As a reality check, when I think back to attending school, in a room
of 30 to 35 kids, it was about as likely as not that two kids would
share the same first name.

A friend of mine worked for a social service organization many years
ago placing children with potential adoptive parents. She was
extrememly meticulous in providing documentation to the adoptive
parents regarding the birth mother's health, family history, and other
things including any names the mother might have mentioned considering
(she then had the documentation secured by an attorney with
instructions for release as dictated by law and the wishes of all
parties).

I wonder if someone involved in the placement process with your mother
might have discussed names with her, even casually, and passed that
info along.

To expand slightly on Ansel001's comment, my fifth grade class had 3
Michaels, 3 Davids, 2 Pats (girls), 2 Johns, and 2 Cindys out of 22
kids.

Finally, I recall a stat that said in a random sample of approximately
35 people, odds were very close to even money that two of them would
share the same birthday. Dunno why I mention it other than it is
pretty neat.

Ansel,

I answered the question under the assumption that ANY pair of names
would have been considered remarkable. If I had calculated the odds
that both names would be Christy, I would have gotten 0.7% times 1.1%
or about one chance in 10,000 instead of about 1 in 100.

Markvmd,

This is exactly analogous to the situation you mentioned about the
coincidence of birthdays in a class of ~30 people. If you ask the
probability that two people will have birthdays on Dec. 25, it is much
smaller than if you ask whether two people will have the same
birthday. The probability of the latter is quite high.

This is why I was careful to be precise about the formulation of the
question. Note that I did not ask: "what is the probability that the
younger sister and older sister will both be named Christy".

markvmd brings up the interesting classical problem of two people
sharing the same birthday.  Actually it takes only 23 people to have a
better than 50% chance that two of them will share the same birthday.
This assumes you don't specify what two people or what birthday.

The other question about shared birthdays is:  How many randomly
selected people does it take to have better than a 50% chance that one
of them shares a birthday with you?  This is counterintuitive in the
opposite direction from the previous question about birthdays.  If my
math is right it would take 253.  This is the expected number of
people it would take for there to be more than (365/2 = 182.5)
different birthdays among them.  By the time you get this many people,
it would be expected that quite a few of them would share birthdays.

My reality check is one of the companies I worked for (with a couple
hundred people) put out a list of everyone by birthdays during the
year.  You could see lots of days in the year with two people sharing
a birthday, several with three, one or two with four, and quite a few
days with no one.

I was very interested in Mark's first comment, wanting to raise the
same point last not, but could not post it:

Can we rule out that your parent who is the parent of Christy 2 did
not meet her adopting parents?  If this might have happened, the
latter might have asked what her name was  - or what the parent wanted
it to be -  and accepted that suggestion.  Obviously many parents
consider choosing one of the then more popular names (Christy, 4th,
then), a name which the adopting parents would be likely to find
acceptable.
That parent might then later have prevailed with the choice of your
sister Christy?s name, still finding it attractive or maybe wishing to
remember the first child.

Regards, Myoarin

Thank you for the responses so far.  Here are my comments and thoughts
based on your comments:

1. I'm realizing that the fact that they are half-sisters is
irrelevant, since they were completely randomly chosen.  In fact, the
birth mother of Christy2 gave Christy2 a different name when she was
born, so neither child-namer had any idea as to what each other's
children were named.

2. I liked the classroom analogy.  A considerable error with that
comparison is that the children in the same classroom would we closer
in age, and thus, would have a higher probability of having the same
name as another student.  In my proposed situation, the separation of
years between the births of Christy and Christy2 offer different name
popularity.

3. Kottekoe-ga brought up the point that we would need to determine
the percentage of births for a given name, rather than just the rank. 
Since I did not offer the real names or years, the percentage could
not have been determined.

Having looked up the percentage of births using each name for the
corresponding years, my findings are as follows:

P(Christy)  = 0.8597% = .008597
P(Christy2) = 1.8762% = .018762

I then see the P(Christy) and P(Christy2) BOTH happening is:

P(Christy) * P(Christy) = .008597 * .018762 = 0.000161296914 ~ 1/6200 chance

According to http://funny2.com/odds.htm that's about the same odds as
injuring yourself while shaving.  According to
http://www.nsc.org/lrs/statinfo/odds.htm that's roughly as probable as
a "Fall from out of or through building or structure."

To give credit, kottekoe-ga, answered the question in his second comment by saying:

"I answered the question under the assumption that ANY pair of names
would have been considered remarkable. If I had calculated the odds
that both names would be Christy, I would have gotten 0.7% times 1.1%
or about one chance in 10,000 instead of about 1 in 100."

Thanks for the input.
Any discrepancies in this reasoning?

Inspectanek,

Actually, the correct way to do the calculation is to calculate the
odds of ANY pair of names, since you would have been equally surprised
to learn that both had the name Kimberly, rather than Christy.

This calculation is much easier in the case of birthdays in the
classroom, since birthdays are uniformly distributed over the year. To
do it precisely for names, one would have to use the probabilities of
the most commonly used names and calculate from there. My argument was
meant to show that the answer is close to 1 in 100.

No, kottekoe-ga, I disagree.  If they were both named the most
unpopular names for their respective years, I would be more suprised. 
This point becomes obvious if you consider the extreme case, in which
all children are given the same name at birth.  I would certainly not
be suprised to find out that two children had the same name.

A friend pointed out that ansel001-ga's comment was valid, with regard
to being equally suprised had my other sister had the same name as my
new sister.  Without doing the math at this moment, I figure it would
cause the formula to be more like:

P(AB) + P(CD), P(AB) is for one sister's name, and P(CD) is for the
other sister's name.

It would decrease the chances a bit.