User blog comment:Linathan/Misunderstandings about Summon Rates in Brave Frontier/@comment-26947994-20180202015540

"we have at least a 76.27% chance of at least one green summon circle appearing out of the five summoning circles we can summon from. Upon summoning a green circle, we have a 3% chance of obtaining Azura. Combining the probabilities together, that's a 2.289% chance of obtaining Azura within the first five summons."

2.289% is a better model for obtaining Azura within the first summon, since we only apply the 3% probability only once to every situation where we get at least 1 green circle. If we include the first five summons, we must consider the situations when there is more than 1 green summon circle. For example, on the 1 in 1024 chance that you get 5 green summon circles, the probability of obtaining Azura (using complementary probability ) is 1 - (0.97^5) = 14.13%, which is much greater than 3%. Then, we multiply this by the 1 in 1024 chance to get this situation, resulting in a 0.0138% chance. Although this probability seems rather underwhelming, they add up over the diffferent scenarios. The individual probabilities of each scenario are below:

5 Green Circles: 1/1024 * (1 - (0.97)^5) = 0.0138%

4 Green Circles: 15/1024 * (1 - (0.97)^4) = 0.1680%

3 Green Circles: 90/1024 * (1 - (0.97)^3) = 0.7675%

2 Green Circles: 270/1024 * (1 - (0.97)^2) = 1.5583%

1 Green Circles: 405/1024 * (1 - (0.97)^1) = 1.1865%

0 Green Circles: 243/1024 * (1 - (0.97)^0) = 0.0000%

Adding up all our probabilities, this gives us a total of 3.6941% probability of obtaining Azura within our first five summons. And so, as Linathan said himself in section 3, "Your math is wrong." :P

But really, I'm too much of a math nerd.