A Is the binomial a special case of the beta binomial?

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The beta-binomial distribution can approximate the binomial distribution very closely when the parameters α and β are large. The term "arbitrarily well" indicates that one can achieve a distribution as close as desired to the binomial by selecting sufficiently large values for these parameters. However, the binomial distribution is not a special case of the beta-binomial distribution, as there are no finite values of α and β that equate the two. Instead, the binomial is considered the limiting case of the beta-binomial as both parameters approach infinity. This distinction is crucial for understanding the relationship between these two distributions.
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On Wikipedia one can read in the article Beta-binomial distribution:

It also approximates the binomial distribution arbitrarily well for large ##\alpha## and ##\beta##.

What is the meaning of 'arbitrarily'?
On Wikipedia one can read in the article Beta-binomial distribution:

> It also approximates the binomial distribution arbitrarily well for
> large ##\alpha## and ##\beta##.

where 'It' refers to the beta-binomial distribution. What does 'arbitrarily well' mean here?
 
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Ad VanderVen said:
What does 'arbitrarily well' mean here?
It means the same here as it always means: you can get as close to the target as you want by aiming carefully.

In this case, you can get a distribution as close as you like to the binomial if you choose large enough ##\alpha## and ##\beta##.

Ad VanderVen said:

Is the binomial a special case of the beta binomial?​

No (because there are no finite values of ##\alpha, \beta## that make them equal), but it is the limiting case as ##\alpha## and ##\beta## tend to infinity.
 
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