According to [Summoning Rates](
https://sskotz.gtarcade.com/en/gameplay/1934.html), "all S Saints have [an] equal probability of acquisition" (2%), while "the probability of the Themed Summon Saint will be higher than the other S Saints in Themed Summon Pool."
We have 17 S Saints in the Advanced Pool and Libra Dohko (04/12/19-11/12/20) in the Themed Summon Pool.
I do not remember where, but I think that I read something about the Themed Summon Saint having the double probability of acquisition than any S Saint. Meaning that the Theme Summon Saint has a chance of 10.53% (2/19 = 0.1053) among the S Saints of the Themed Summon Pool.
Combining all those things mentioned above, we have that the probability of obtaining the Themed Summon Saint is 0.21% (2/100 * 2/19 = 0.0021) for each Advanced Astral Gem spent.
Anybody would want to know how many gems are needed to obtain the Themed Summon Saint or what is the probability of not obtaining the "banner" even though a certain number of gems were spent.
We already know that approximately two of a thousand players will get the banner spending just one gem (0.21%). Assuming that each summon is unaffected, in terms of probability, by whatever it happened in the previous summons. The probability of obtaining the "banner" exactly with the 100th gem spent is 0.17%. Using this way of obtaining probabilities, we have that 34.39% of the players will obtain the "banner" with the first, the second, ..., or the 200th gem, i.e. 34.39% of the players will spend 200 gems or less to obtain the "banner"; 22.56% of the players will obtain the "banner" spending 201 to 400 consecutive gems; 14.80% of the players will obtain the "banner" spending 401 to 600 gems; And, 28.25% will need 601 or more gems.
On the other hand, suppose that I am going to spend a specific amount of gems no matter what, no matter how many "banner" saints I get or do not get. Again, assuming independent summons, i.e., probabilities unaffected by previous summons. The probability of not obtaining the "banner" after spending exactly 100 gems is 81.00%, i.e., 81 of 100 times that I do this, I am not going to obtain the "banner". Spending 200 gems, the probability of not getting the Theme Saint is reduced to 65.61%. Spending 329, we have approximately a fifty-fifty chance (49.99% of not getting the "banner").