What is the probability of four people standing back to back in line at a restaurant having the same, common, first name?

Question

I'm trying to calculate the odds of this happening because myself and three other men named Steve ended up in line together at a Taco Bell. It seems very strange but maybe it's not as strange as I think. I did research weighted probabilities and I don't quite understand how to calculate this.

Answer 1

To have four in a row with the same name, the first one can be any name and the other three have to match it. Make a list of all the first names and say the probability of name $i$ is $p_i$. If the first person has name $i$ the chance the other three also do is $p_i^3$ so the chance is the sum of this over all the names $$\sum_ip_i^3$$ This assumes the people who visit Taco Bell are representative of the general population. Otherwise you can take your $p_i$ to be the proportion of Taco Bell customers with name $i$. Wolfram Alpha will help with the data, but it may be hard to interpret. This says if Taco Bell customers are all $58$ years old and born in the US the chance that four men in line will all be specifically Steve is about $\frac 1{16}$ because half the boys born in $1960$ were named Steve. I find that fraction hard to believe. There were a lot of Steves in my classes, but not a third of the boys. I would say it was closer to $10\%$ even if you count all forms of the name.

Answer 2

Accoring to Wolfram Apha, 0.475% of the US population is named "Steve" or "Stephen" or "Steven."
http://www.wolframalpha.com/input/?i=stephen

Suggesting that in any given line, you have a $4\cdot (0.00475)^3$ chance of being part of a "quartet of Steves."

How freakish is this? over a lifetime, how many lines have you stood in? And how many did you ask the names of those that stood around you?

If we look at all names, then we need to make some assumption of the distribution of names, and the lengths of lines (how many groups of 4 can we select from a given line) the problem becomes more complicated.