Facebook maths question

[Grantley]

Is it mathematically probable (>50%) that a majority of the Facebook friends (>50%) of any randomly selected subject will have a greater number of Facebook friends than the subject?

For example, pretend there are only 10 people on Facebook: Andrew, Barry, Charlie, Dave, Edgar, Frank, Gary, Hank, Ian and Jonesy.

The boys have mixed popularity. Andrew is friends with Charlie, Edgar, Hank and Ian so A(CEHI).

A(CEHI)
B(E)
C(ADE)
D(ECFJ)
E(ABCDHJ)
F(DHI)
G(J)
H(FAE)
I(FA)
J(GED)

At random, let's grab Charlie. Charlie has three friends, but more than half of his friends have more friends than he does. The same is true of five of the others, so 60% of the boys have friends with more friends than they have.

If I fiddle with the relationships (within reason) it doesn't seem to affect the 60% statistic very often. I think that's because any given person is more likely to be friends with a popular person than an unpopular person.

What do you think?

[The Gameboy]

..... What?

[Larynx]

Sure, why not...?

[Porcell]

I'm not sure that the problem is solvable as stated with probability. Now if you had a known population and a known data set, you could answer the question for that data set, but you can't make a judgement on a random individual because 1) you have no way of predicting how many friends that person is likely to have and 2) you have no way of predicting how many friend his friends are likely to have.

A known data set would look like this:

Adam: 100 friends. 40 of those 100 friends have greater than 100 friends themselves. [false]
Barney: 250 friends. 180 of those 250 friends have greater than 250 friends themselves. [true]
Charlie: 40 friends. 40 of those 40 friends have greater than 40 friends themselves. [true]
Dean: 300 friends. 100 of those 300 friends have greater than 300 friends themselves. [false]
Edward: 500 friends. 300 of those 500 friends have greater than 500 friends themselves. [true]
Frank: 700 friends. 400 of those 700 friends have greater than 700 friends themselves. [true]

So of these 6 people, you have a 4 in 6 chance of randomly selecting a subject who has a majority of friends with more friends than the subject has.

You would have to boil each person down to one thing; do more than half of that person's friends have more friends than the original person. Yes or no. For me, I have 187 friends. I would have to go through every single one of those 187 friends and see if they have Greater or Less than 187 friends themselves. Once I know those 187 pieces of information, I can determine whether 94 or more friends (50%) have more than 187 or not, and thus label myself as a True (the majority of my friends have more friends than I do) or False (I have more friends than the majority of my friends)

[Windry]

Feels like a trick question. Without any actual data from Facebook, it'd be impossible to tell.

I can think of 2 simple possibilities for both.

<50%
A,B,C,D,E,F are on facebook.

A: E and F
B: E and F
C: E and F
D: E and F
E: A,B,C,D,F
F: A,B,C,D,E

>50%
A: C,D,E,F
B: C,D,E,F
C: A,B,D,E,F
D: A,B,C,E,F
E: A,B,C,D,F
F: A,B,C,D,E

If you assume a perfect bell curve for the placement of people with high # of friends and low number of friends, it would seem like the possibility would be exactly 50%. Of course there's no way the real life placement is like that, but without any stats from Facebook, how would you know?

[eriseis]

Charlie has three friends, but more than half of his friends have more friends than he does. The same is true of five of the others, so 60% of the boys have friends with more friends than they have.

It is known in the social networking field that if someone doesn't break 10 friends on a social networking platform they're exponentially much more bound to leave than someone with more than 10 friends. If Charlie has 3 friends 1) he does not care about Facebook and 2) everyone else has more friends than him :P

[Porcell]

It is known in the social networking field that if someone doesn't break 10 friends on a social networking platform they're exponentially much more bound to leave than someone with more than 10 friends. If Charlie has 3 friends 1) he does not care about Facebook and 2) everyone else has more friends than him :P

Or it's a god damn example in this hypothetical question...

[fonzatron2000]

https://www.youtube.com/watch?v=5rax50efCUk

[Meteoric]

Intuitively, I strongly want to say the odds should be very near 50%, and tend to get closer to that with larger sample sizes, because it seems like you should be able to relate this to an average value somehow. But I'm not sure that will actually be true. It's worth noting that, because of the small numbers in your example, you might have more people with exactly the same number of friends than a real analysis of something like Facebook would probably have.

This also seems like it should have some connections to graph theory, if you want to look further into it.

[youngfool]

my mind exploded

[Badpaladin]

Yeah, can't answer that without any kind of data. If I were to guess, it's not true because there are probably many more people with profiles who barely use Facebook than those who have tons of friends. I don't know the statistics of median/mean friends per person, though.

[Taliesyn]

Basically, what you're asking is if the mean number of Facebook friends is higher than the median number of Facebook friends. Without specific data on the number of friends that Facebook users have, including mean, median, and distribution, the question cannot be answered.