According to wikipedia the most common first names for males in the US according to the 1990 census are, in that order:
Let's use an approximate probability distribution (any better way?) that gives the probabilities to these names according to their order i (starting at 0) following:
P(name_i) = 0.15 / (i + 2)
This is a probability distribution for at least 1000 names (sums to 1), and starts like this:
0.0749, 0.0499, 0.0374, 0.0299, 0.0249, ...
meaning that someone has 7.49% chances of being called James. The sum of the probabilities of having a name starting with J from these list is 14% (P(James)+P(John)+P(Joseph)), lets call it Pj. Of course Pj is in fact higher because more names than these 8 in the list of 1000 may start with J.
Now let's assume a child is given three first names, the probability that at least of them starts with J given pj is 36%!! (from (1-(1-pj)^3)). So every time you meet a new person, you have one third chances to see him having a J on his business card... that's huge. Let's call this probability pIJ, probability of at least one initial starting with a J.
However, when you say overwhelming majority, let's assume that you mean 8 out of 10. The probability that out of 10 people you meet in a row 8 have a name starting with a J is, using binomial distribution, only 0.5% I'm afraid, by comb(10,8)pIj^8(1-pIj)^2 .
However, if you settle with 5 our of 10, your chances rise to 16%.
Please someone correct me if the math is wrong.