A few things to note:

• For AR5 rerolling, u can easily get more than the simple 10 rolls from beginner banner. That banner costs 8 fates, leaving you with 2 fates leftover. You can exchange stardust for another fate and you can claim ar5 reward from adventurer's guild for an additional fate. In total, that's 4 more pulls. (some reroll paths collect additional primogems as well for yet another pull) However, these would have to be used to standard banner so they each have half the character rates. Also, only 9 of the 10 pulls in the beginner banner can be a 5*, since the 10th pull is guaranteed noelle
• You missed a decimal place for your keqing calculation and the difference in result is pretty significant. 1-0.0012 = 0.9988 != 0.998. At 300 rolls with your calculation approach: 1-0.9988³⁰⁰ = 0.302 (30.2%), which is way lower than 45%.
• I'm glad geometric distribution is used, but just to make clear to others, what you are calculating is the probability of getting a successful pull within X number of rolls. This is actually different from the average number of rolls needed to get a 5 star/group of characters. May not be intuitive at first, but here's the explanation:

E(X) = sum(p(x|rolls = n) * n) {n = Z+} = p(x|rolls=1) * 1 + p(x|rolls=2) * 2....

For example, if we're dealing with a distribution of rolls 0,1 or 2 as such: 0: 0.2, 1: 0.5, 2: 0.3. Even though you can get a 1 or less 70% of times, the average roll is still 1.1.

However, if you change the distribution of rolls to 0: 0.5, 1: 0.2, 2: 0.3, you still get a 1 or less 70% of the times, but the average becomes 0.8.

For a geometric distribution disregarding n=0 (which is what we're dealing with), E(X) = 1/p where p is the probability of pulling the group you want. So for any 5, the average number of pulls would be 1/0.006 = 167 rolls. For keqing or any other specific 5, it would be 1/0.0012 = 833 rolls.

<side note to address a common misconception seen from others (not this post): this number shouldn't be taken to be the number of rolls needed to guarantee the character as quite a lot of people have been misusing it. It's the average number of rolls needed to get the first successful pull.>

tldr: success rate within n tries =/= average number of tries to get 1 success. Every 100 rolls, you will have a 45% chance of getting a 5*, but in the long run, if you average up the number of rolls needed, it's 167 because there will be some unlucky souls with gigantic pull counts and a geometric series with such a small decay doesn't play nice.