When two people talk, and they have different native languages, what is the chance that they speak in English?

How I got there:

• start by taking the "L2 speakers" column from wikipedia https://en.wikipedia.org/wiki/List_of_languages_by_total_number_of_speakers
• assume that every "conversation between two people with different native languages" is between two people selected at random. So for each language, we square the number of L2 speakers to get the number of possible pairs of people that could converse in this L2 language.
• divide by the sum of the squares to normalize

e.g. the two largest listed are English at 1 billion L2 speakers and Modern Standard Arabic at 275 million L2 speakers. So, for English, there are 1 billion * 1 billion = 10^18 possible conversational pairs possible, while for Modern Standard Arabic there are 275 million * 275 million = 7.5*10^16 pairs.

Note that this means that conversations are much more skewed towards the most-spoken languages than the raw numbers of speakers.

Does anyone have an alternate/better approach? E.g. I think naive squaring is probably too simplistic, since people will tend to cluster geographically by their L2 languages.

Note: not sure if this is the ideal forum for this question, but it seems similar in spirit to some existing questions like What languages together maximize the number of people you can speak with?

• Well, for one thing, you’re ignoring L1 speakers. Consider a case like Mandarin. To begin with, the numbers seem low for that one – a sum total of 1.138bn speakers when Mandarin is pretty universally spoken by ~1.5bn people in China and a fair part of the Chinese diaspora around the world doesn’t add up (unless Ethnologue is excluding L3, L4, L5, etc. speakers). But even with just 939m L1 speakers and 199m L2 speakers, that’s (939m * 199m = 1.86 * 10¹⁷) + (199m * 199m = 3.96 * 10¹⁶) = 2.26 * 10¹⁷, an order of magnitude(ish) more than your figure for Arabic. Commented Feb 7 at 2:53
• I don't think there is any way to know the chance of any two speakers (L1a and L1b, to distinguish two native languages) who don't know each other's language speaking some third language (L3) to each other. Let's say you are in West Africa, where two people speak two different local languages, the chance they would speak English to each other is probably much less than their speaking French to each other. But even so, there could be some third local language they can speak to each other in and not English or French. Commented Feb 7 at 14:50
• @JanusBahsJacquet Sorry, my framing was misleading, I was intentionally excluding L1 speakers (i.e. both people involved are speaking non-mother-tongue). Actually, both versions are independently interesting (at least one person speaking non-mother-tongue, and both people speaking non-mother-tongue).
– user43398
Commented Feb 10 at 17:56
• @Lambie of course there is no way to know for sure, which is why I'm trying to estimate. But consider that e.g. in the year of 2023 (for concreteness), there was some specific, finite number of such conversations that happened, which could in theory have been counted. So the number is unknown but not fundamentally unknowable.
– user43398
Commented Feb 10 at 17:59
• Two people at random don't necessarily speak the same L2 or even speak an L2 at all. I know every little about statistics and math, but I do know these calculations won't mean much. The numbers are finite, they are not definable but the categories are not. Commented Feb 10 at 18:39

Roughly 3% probability, if we just consider random meetings of any two people on earth, and if my math is right.

There are roughly 8 billion people, 0.5 billion native speakers of English (`1/16`) and 1.5 billion total speakers of English (`3/16`).

So if 2 people are selected at random:

• Upper bound: 3.5% chance that both can speak English (`(3/16)²`)
• Lower bound: 0.4% chance that both are native English speakers (`(1/16)²`)

It will be less than 3.5% because many people who can speak English still speak another language to each other, especially if they share another mother tongue. And it will definitely be more than 0.4%.

That said, the most likely scenario is simply that they have no common language of communication.

• But that's the chance that any two people out of the total world population can both speak English. The situation here is that two people who aren't NSs of the same language are already talking to each other. So you need to remove all the monolingual people from the 8 million, for a start. And there's no chance that they have no common language of communication, given they're already communicating in a language! Commented Feb 13 at 11:16
• @Araucaria-him Sure, if that's what the OP was envisioning - not totally clear to me. I was mainly trying to help with avoiding making the math too complicated. Commented Feb 13 at 12:37
• @Araucaria-him The OP is saying that L1 and L2 speakers cluster geographically. That's really a no-go to start. Then, If I am Portuguese L1 speaker and am speaking to a French L1 speaker in English, how can I know how many of my fellow countrypeople - or theirs for that matter - can do that same? Commented Feb 13 at 14:55