Chance on Borders between 20 Random Countries Given 1 Country

If we pick 20 random countries from the Countries of the World we have a 95% chance of a border between them. Does this chance change if we fix one of the countries and randomly select the other 19?
Since this is dependent on the number of borders of the country, each country is plotted by its number of borders, not by its name.
The data was generated using 12 runs of 1,000,000 random sets of 20 countries, and analysing which of these sets had borders in them. This is why each country has 12 dots.
Saudi Arabia stands out since it borders 4 countries with 1 or 2 borders, which are normally more difficult to create borders with.
The percentages depend on the random sets of 20 countries, and can thus change a bit between runs and countries even if they should be the same.
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