The probability that seven people have different first initials

Last week I looked at some data collected by Rick Wicklin to determine if first and last initials are independent. (Conclusion: no.) This brought to mind a question closer to home. Literally. I live in a house with seven people. We all have different first initials. What’s the probability of that? (This is good to know if, say, you want to write a single initial on your food and assure it doesn’t get eaten; this might be the context in which I noticed it. Although I may have noticed it because we have a spreadsheet that magically keeps track of household expenses, I don’t remember.)

The quick-and-dirty way to solve this is, of course, simulation. The distribution of first initials from Wicklin’s data can be found in R. Assuming the matrix of counts by first and last initial is x, we can get the vector of frequencies of first initials
firsts = rep(0,26); for(i in 1:26){firsts[i]=sum(x[i,1:26])}
and we can extract the distribution explicitly:

j m s d c b k a r l t p e g w n h v f y i o z x u q
579 403 390 387 342 306 294 289 264 240 230 166 134 125 75 62 61 42 40 23 15 11 10 7 5 2

I won’t explicitly use the distribution, but if you’re curious: R has a built-in vector letters which contains the letters of the alphabet. letters[order(-firsts)] puts the letters in the order coming from sorting the frequencies of first initials in descending order; that gives the first row. The second row is just sort(firsts, T). I follow Howard Wainer’s dictum of not listing in alphabetical order, a practice he memorably calls “Alabama first”.

Then to take a sample of size 7 with replacement from this distribution – to simulate my house – we can run
sample(1:26, 7, replace=TRUE, prob=freqs)
where I’m just sampling from the vector 1 to 26 because it’s easier that way. Some samples (the first six off the presses) are

16 11 18 10 7 10 10
13 18 3 7 18 13 1
10 12 3 4 4 23 13
8 6 13 19 11 4 4
13 3 2 20 14 3 2
5 1 10 1 26 19 20

which correspond to the septuples of initials PKRJGJJ, MRCGRMA, JLCDDWM, HFMSKDD, MCBTNCB, EAJAZST. In particular each of these has at least one repeated initial, so we start to get the sense that seven people chosen at random having all different initials is relatively rare.
To get the length of such a sample we can call it s and run length(table(s))table generates a frequency table, and s gives its length. (This may or may not be the fastest way.)

So the single line of R (alright, for an obnoxiously literal definition of “line”)
x = rep(0,7); for(s in 1:10^6){i = length(table(sample(1:26, 7, replace=T, prob=freqs))); x[i]=x[i]+1}
gives the frequency table of the number of different first initials in samples of size 7, over a million simulations. The resulting table is

number of different birthdays 1 2 3 4 5 6 7
number of runs 2 213 7636 80396 300451 424405 186897

In particular the probability of having seven different first initials is around 0.187. In comparison, the “traditional” birthday problem (where all birthdays are equally likely… well, except for February 29, which I noticed earlier this week when I taught the birthday problem on February 29) gives us that the probability of seven people having all different initials is

{(26)(25)(24)(23)(22)(21)(20) \over 26^7} = 0.41277

and so collisions really do become much less likely. A collision among seven people is about as likely as it would be if there were fifteen possible different initials, all equally likely. The probability of this is (15!)/(8! \times 15^7) \approx 0.1898. And there are other senses in which there are “effectively” only about fifteen possible first initials. For example, if $p_i$ is the probability that a random person’s first initial is the $i$th letter, then $(\sum_i p_i^2)^{-1} \approx 0.0710$; this number, roughly 1/14, is the probability that two people chosen at random from the population has the same birthday.

In fact, a naive approximation to the probability that no two of these seven people have the same first initial comes as follows: there are {7 \choose 2} = 21 pairs of people. Each pair has probability $\sum_i p_i^2 = 0.0710$ of coincidence. So the expected number of coincidences is 21(0.0710) = 1.491. If we assume that the distribution of the number of coincidences is Poisson (which it’s not!), the probability of at least one coincidence is exp(-1.491) = 0.225. Not bad. (The same model gives e^{-21/26} \approx 0.446 for the uniform distribution, where the correct answer is 0.413.

Alternatively, the Shannon entropy of the distribution of first initials is $2.80$, which is the same as the Shannon entropy of a uniform distribution on a set of size $e^{2.80} \approx 16.4$. (You know, if such a set existed.)

30 thoughts on “The probability that seven people have different first initials

  1. Munford (TAS, 1977) showed that ANY deviation from uniform frequencies increases the probability of a match (or, in your language, decreases the probability of no match). This is not very noticeable for the standard Birthday Problem because even though the empirical distribution of birthdays is not uniform (see, the deviation from uniformity is relatively small. Consequently, the empirical probabilities of a match are not very different from the probabilities obtained by assuming a uniform distribution. As you’ve noticed, however, the distribution of initials is FAR from uniform, so there is a big difference between empirical estimates (obtained via simulation) and the exact probabilites assuming uniformity.

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