Card Packs are the standard TCG method of adding cards to one's collection. In Artifact, each $2 pack earns the owner 12 random cards upon opening. Packs are also offered as rewards for leveling up your account, and for winning in various Expert Play modes.
All packs have a standard composition as follows:
In addition, each Common card has a 5% chance to be upgraded to an Uncommon card, and then each Uncommon card has a 5% chance to be upgraded into a Rare card. These chances for each card are separate; it is possible to have multiple Rare or Uncommon upgrade procs. These chances are also cumulative; it is possible (although unlikely) for a Common card to be upgraded to an Uncommon, and then for that Uncommon to be upgraded to a Rare.
When generating the cards that are contained within a pack, Artifact first determines which slots will contain Hero and Item cards, and then separately determines the rarity of each slot afterwards. This means that Uncommon/Rare Item and Hero cards are somewhat more rare than one would assume based on their quantity.
For example, in the Call to Arms set, there are 77 rares, 12 of which are Heroes. This means 15.58% of all Rares are Heroes, but because of the ordering mentioned above, the Rares in any given pack have only a 8.33% chance to be Heroes. The discrepancy is much less pronounced for Items due to there being two slots in every pack: Items make up 16.88% of all Call to Arms Rares, but pack Rares have a slightly diminished 16.66% chance of being an Item.
Because of the upgrade chance, each pack has on average 1.17 Rare cards.
Additional probability information (as well as empiric data) can be found in this Reddit thread here.
All card packs are priced at $1.99 USD. Purchasing packs in larger quantities does not provide a discount; this is only included for convenience.
All new accounts can earn up to 15 booster packs by leveling up via gameplay; levels 2 and 4 award 1 card pack and 1 Event Ticket, levels 6, 8, 12, and 14 award 2 packs and tickets, and level 16 awards 3 packs and tickets. There are no other means of grinding out card packs outside of the Expert Play methods listed above.