Pronunciation of "priv-" in British English and American English

For example, the pronunciation of "priv-" in the words privacy and private is different in British English. The former is pronounced as prɪv- whereas the latter as praɪv-. Yet, in the US, the pronunciation is the same.

Meanwhile, the word privilege is, so far as I know, pronounced the same in both British and American English (prɪv-).

There seems to be something very arbitrary in all this. But there must be an explanation. Any ideas?

The English language is full of inconsistencies, generally, and each of the variants (British vs American for example, or 19th century vs 21st century) may add further inconsistencies or remove them. In that sense, it is fundamentally quite arbitrary.

In each instance of this sort of thing one could probably uncover the history behind it, and that can be an interesting exercise (lots of which happens on this forum). But, my broad advice would be "Just accept it and get on with your life".