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One of the main features of Wizards is that they can have vast numbers of spells in their Spell book.

I have not played in Adventure League games, nor run any of the Pre-built adventures, so have no experience of how frequently this may have been designed to happen.

Is it assumed that Wizard characters will find new spells during their adventures, or is gaining two per level seen as normal, and its a feature that they can get extra spells...

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Based on my experience, it's pretty rare to find a usable scroll. I blame AL rules and/or DM obliviousness on the lack of new spells available to wizards. I have had luck in trading with other wizard PCs for spells, using gold and downtime to copy their spells into his book.

as a result, my 3rd level wizard has about as many 1st level spells as are available. And exactly 2 2nd level spells...

However, based off of the rules, yes, it is assumed that a wizard expands their spell repertoire by finding scrolls / spell books while adventuring.

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Yes.

The PHB says:

You might find other spells during your adventures. You could discover a spell recorded on a scroll in an evil wizard's chest, for example, or in a dusty tome in an ancient library.

Additionally:

Spell scrolls are listed on the random treasure tables in the DMG and the Monster Manual mentions that some monsters, such as a Lich may have "libraries of spellbooks".

It is also possible to simply buy them. (For a value of "simply" that depends on how common wizards are in your setting).

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    \$\begingroup\$ I think the question was asking whether it's the default assumption that wizards will find spells during adventures, not whether it's a thing that can happen. \$\endgroup\$ – user2357112 Jan 15 '17 at 21:02
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    \$\begingroup\$ @user2357112 — Being listed on the default random treasure tables is probably the strongest bit of evidence in my answer to support my assertion. \$\endgroup\$ – Quentin Jan 15 '17 at 21:04
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    \$\begingroup\$ @Quentin Looking in the DMG it gives number's for how many times a party gets a Treasure Hoard, and doing the math on the tables. I calculate that of the seven rolls on the CR 0-4 table that only equates to 2.1 scrolls received. maybe 50/50 chance being a wizard spell. So by my maths you are only likely to find one scroll in the first 4 levels. Does this seem about right? \$\endgroup\$ – Whinja Jan 15 '17 at 22:45

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