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The title might be a bit too specific; here is a more detailed explanation of the question:

While picking a new language to learn, it is very often a decisive factor how this new language helps one to communicate with more people than without it. If a major language is very prevalent as second language in a particular country, it often makes no sense (e.g. as a tourist with limited time in the country) to learn the official/most common language - the benefit of learning the local language in terms of how many people one can communicate with is negligible. In such a case it is more reasonable to for example only learn Russian when planning to go to Ukraine, Kyrgyzstan, Belarus or Kazakhstan, Arabic or French for Morocco and Algeria or English generally everywhere.

Let's say I am fluent in my native language (L1) and two other languages (L2) and I would like to estimate with how many people in the world I can communicate. A straight-forward approach could be to sum up the total number of speakers of each language (L1+L2). As an example, using the data from the Ethnologue (2019), we would get a total of around 1.925 billion people with the language combination English+Spanish+Russian.

But this overestimates the number of unique people I could speak with. In fact, people who are for example L2 speakers of English might already be included in the sum as L1 speakers of Spanish or Russian, and vice versa. The correct number of unique people I could speak with must be somewhere between the sum of L1 speakers (992.8 million) and L1+L2 speakers (1924.5 million) - that's a difference of almost a billion people!

Language | L1 (mil.) | L2 (mil.) | L1+L2 (mil.)
English  | 379.0     | 753.3     | 1132.3
...      |           |           |
Spanish  | 460.1     | 74.2      | 534.3
...      |           |           |
Russian  | 153.7     | 104.4     | 258.2

How can the correlation between L1 and L2 speakers of different languages be estimated? If I were to correct the sum from above by counting each person that speaks either English, Spanish or Russian only once, what effect would this have on the sum?

As a further example, assuming I already speak English and German, the total number of people I can talk to would not increase appreciably if I decided to learn Dutch, since most Dutch L1 speakers already speak at least one of those as second language.

Ideally, I am looking for something like a matrix/table that shows how many L1 speakers of each (major) language are also L2 speakers in another language. From this it would be possible to exclude people that would otherwise be counted twice. I'm not sure if this has been studied or has a name. I couldn't find any paper, article or book on this, so a source might also be very helpful.

With regards to the comment by Tommi Brander, I would like to find out or get an estimate for the union |X ∪ Y ∪ Z|, where X, Y and Z are the total number of speakers of three languages. In reality it looks probably something like this (areas of the circles are proportional to the total number of speakers, intersections are just guessed): enter image description here I guess it's safe to assume that every person only has one native language, and for the sake of simplicity also only max. one second language.

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    A relevant mathematical tool: en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle – Tommi Brander Mar 11 at 14:49
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    Also, most of the languages of India are probably out, since although many of them have very high numbers of speakers, English language education is extremely common there. My WAG would be English, Russian, and Arabic, with a possible inclusion of French or Spanish, but I'll leave an actual answer to those with more proficiency in the tools. French is a very common second language in many African countries, but then many people there also study English and/or Arabic too, possibly making French superfluous. – Robert Columbia Mar 11 at 15:08
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    The least overlap of the major languages will likely be English, Chinese, and Arabic. The other two languages are not spoken widely by any of these three. These are three of the top six languages, if you count Modern Standard Arabic as the relevant Arabic variant (even though it is no one's native language). – AML Mar 14 at 1:58
  • @AML Yeah, that sounds reasonable, or maybe something like Chinese, Arabic and Spanish, since English is probably a very popular second language in any area of the world, while Spanish isn't necessarily. – ahemmetter Mar 14 at 11:51

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