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Do visitors see the same version of the page under test every time they access the site?

Yes. You want to offer a consistent user experience to your visitors. If they see a different page/post every time they access your site they would quickly realize they are being your “guinea pigs” and this could influence their behavior.

Moreover, from a statistical point of view, the validity of the test depends on keeping the two “samples” (i.e. the two or more groups of users under test, each one seeing a different version of the post/page) independent.