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The Birthday Problem on Facebook
Not too long ago somebody mentioned ‘the birthday problem’ to me, which is something they’d read about in a book. For those of you not in the know, the birthday problem deals with the statistics of people having the same birthday and the increasing odds with an increased number of people. The shocking fact that brought the conversation up in the first place is that if you have a group of 23 people, the odds that two of them share a birthday are over 50 percent (50.7297234323985% if you want to be precise). This seems far too high at a first glance, so I thought I’d take a quick look at why this is the case.
Taking the problem at its simplest level it’s easiest to look at the chances that people do not share a birthday. The first person for example:
P(n) = (1 – ((n-1)/365)
Taking the problem at its simplest level it’s easiest to look at the chances that people do not share a birthday. The first person for example:
P(n) = (1 – ((n-1)/365)
So since this is the first person n is equal to 1 so we get:
P(1) = 1 – 0 = 1
In percentages of course, that 1 is 100%, meaning there is a 100% chance person one won’t share a birthday with anybody because he has no other people to share it with.
To follow on from this, when looking at two people, you need to factor in all of the chances that have come before, so you end up with a term in the equation for every person;
P(n) = (1 – (0/365) x (1 – (1/365) x … x (1 – ((n-1)/365)
By then taking the chance that somebody won’t share a birthday, we simply invert this by taking the chance that they won’t away from 100%. Doing all this by hand of course is very painstaking, so we employ the use of some computer software, in my case here, Excel.
My real interest came up about this when I thought about Facebook and people sharing birthdays. Plucked from the Facebook website, we can see that the average user (when I wrote this at least) has around 130 friends. If we take a look at the statistics that tie into this on this graph I’ve made we can see that statistically almost everybody should know two people who share a birthday. The actual odds are around 99.9999999994% (12 S.F.). This did surprise me, since I don’t think I could recall two of my own friends sharing a birthday (though that’s probably to do with me being a terrible friend and not remembering).
Comparatively, where I believe the confusion begins is getting these stats mixed up with the stats of a person sharing a birthday with you, which is generally more memorable for obvious reasons. The odds of this are far lower; just a 1 in 365 for two people. At 130 friends, your odds are better at 29.81%, but somewhat lacking from the ~100% of one pair of those sharing a birthday.
In fact even at 365 friends, there’s only a 63% chance of sharing a birthday with somebody else.
I mention this just to highlight the importance of understanding what you’re looking at in statistics. From a simple rewording; “The chances that your friends will have the same birthday” to “The chances that your friends will have the same birthday as you” the results differ considerably. So make sure you take a second to think about what you’re looking at when you see statistics, regardless of the source.
