If an item is rarer than others, is it that much more valuable? Say I am selling a pack of trading cards, and there are 5 cards inside. Four of these cards are basic cards, but the last card has a 1:10 chance of being a special insert card.

Does the fact that the special insert has a 1:10 chance of appearing mean that it is ten times as valuable as a base card?

I understand that value is almost entirely subjective, so I suppose this is a philosophical question more than anything else, but I'm also not really an economist so I don't know if there are any axioms around this.
(1) What value an item has to you depends on your preferences, not scarcity.

(2) The market clearing price of an item will depend on the preferences of market participants and the supply schedule ("scarcity") of the item.

The way you seem to be thinking about value is very "non-economic", if that's a word.