As pointed out in the FAQ on the community cards and based on the research of the fellow steam users, the only factor that will increase your chances of getting a booster card drop is steam profile level.
The bigger the number, the bigger the chances. An extract from the FAQ:
Once eligible, your Steam Level increases your rate of receiving a booster pack drop:
- Level 10: +20% increase in your drop rate
- Level 20: +40% increase in your drop rate
- Level 30: +60% increase in your drop rate
- Level 40: +80% increase in your drop rate
- Level 50: +100% increase in your drop rate (i.e. the rate has doubled)
Here is a similar question asked on steam community.
I will give an extract from that discussion that serves our purpose (post):
Based on Valve comments on booster packs:
Having more eligible games = higher chance of getting a booster pack.
Not getting a booster pack in a while = higher chance of getting a booster.
Yes, over time you will end up with more boosters packs if you have more eligible games. It's all set up so that you are better off to have all your games on one account, instead of spreading them between accounts.
So this is generating some kind of confusion around this subject, since valve is not giving us something clear that we can rely on, and since nobody can run a test/experiment to prove something simply because the system cannot be reproduced or isolated in such way that it can lead to facts.
I think that the three system versions you came up with are purely theoretical and we cannot guess how the selection is being made. Consider the fact that it still gives booster to super low level users that are eligible also to super high level users that have greater chances.
If it really matters the number of games you are eligible for than it could be very likely that the system is picking games randomly and then selects the users based on their chances due to steam level profile to give away the booster.
If it would pick the users first and the games after, the idea of having a bonus if you are eligible for more games becomes useless since it wouldn't rely on the game at all.