What are the chances of crafting a Rare or Uncommon background?

I did some sleuthing on the steam market, and based on the listings, there's consistently about a 10% chance to get an uncommon item, and a 5% chance to get a rare item (85% for common items). There's no 100% hard data on the actual odds, but I'd say the market's the closest we can get.

Note that you CAN obtain uncommon/rare backgrounds and emoticons from non-foil badge crafts. Foil badges actually have nothing to do with the items you'll get - it's treated the exact same as any other badge craft.

Let us consider emoticons obtained during the Summer Sale in 2018.

These are not sought-after items: they are very cheap, and people usually try to get rid of them, no matter their rarity. The price is the same for every rarity, because the supply greatly exceeds the demand. So, we could expect the proportions of each rarity among market listings to mirror the proportions of each rarity among items in circulation.

The proportions are:

• 64% for common emoticons,
• 23% for uncommon emoticons,
• 13% for rare emoticons.

Similarly, for profile backgrounds, the proportions are:

• 69% for common profile backgrounds,
• 19% for uncommon profile backgrounds,
• 12% for rare profile backgrounds.