Saint Quartz Results: A discussion of Saint Quartz to servants gained

So, running through the simulator… I have a question. In game, how many Saint Quartz does it usually take before a SR or higher servant appears, spook or no?

This is mostly cause the simulator would suggest I would be flooded with servants if I had 300 Saint Quartz to spend. I got 2 Illya’s in a single ten-roll on her banner, for example. And It only cost 210, plus gave me a MLB Saphire Magical Girl CE, and 4 Kill on Sight.

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Specific, math-oriented answer: someone on this forum gave a very detailed explanation on probability in FGO gacha rates right here:

Generic answer: You might have just been very lucky in the summoning simulator. Getting a super lucky streak is not unheard of even in FGO (just take a look in any of the roll threads).

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It’s really random, during summer I got 1 SR from 2 10 rolls when it started (SFran), 1 from a ticket in SNito rate up, and two from SNobu rate up (her and Rama). Other times it takes several to get a single 4*

It’s all about chance. For example. For the maid alter banner. I spent 1035 quartz. The first 180 quartz I got 3 maid alters,1 Helena, and 1 mama. The next 855 quartz I got 5 Helena’s, 3 mama’s, 1 drake, and 1 wu.

Odds of gachas are really just throw 30 saint quartz in and pray.

… Went back to the assassin. Single Ticket, Copy of Nightless Assassin.

39 more tickets, nothing, though Saint’s Vessel makes an appearance.

I mostly want to hear others experiences.

Nother 90 Quartz, Jack again. Why Jack?

Like others, it is just luck of the gacha. I have gotten multiple 5*s with only 60quartz and other times it takes me 600 for just one.

https://shipfu.moe/fgo/

For the more mathematically inclined, the simple formula is:
1 - (odds of not pulling servant)^(quartz/3) , with quartz/3 = # of rolls

so for any 5* this is
1 - .99^(quartz/3)

At 300 quartz, or 100 rolls, you have a 63.4% chance of pulling any 5*.

For any 4*, this is
1 - .97^(quartz/3)

At 300 quartz, or 100 rolls, you have a 95.2% chance of pulling any 4*.

For any 5 * rate-up specifically (solo banner), it’s
1-.993^(quartz/3)

At 300 quartz, or 100 rolls, you have a 50.46% chance of pulling the solo rate-up 5*. You have a 25.95% chance of getting spooked (1-.997^100)

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Obviously, be wary of the Gambler’s Fallacy. FGO doesn’t keep account of all the rolls that you’ve done, so if you’re at your 100th roll, and you haven’t gotten that 5* yet, your chance isn’t 63.4%. It’s still that really low 1%.

It’s really a wonder any of us get 5* servants ever, since the rates are so low.

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Well your example can work both ways, if you can bust at your 100th roll because the odds are the same then you can also succeed at your 1st, still I would not get surgery for those odds, on the other hand a repetitive roll for some bits and pixels is another matter assuming the system is fair.

It’s very random, I remember during Merlin that I needed 5K quartz to obtain ONE 5* servant and it even wasn’t him and three other 4*. Then I proceeded to summon him with my 300 SQs and 10 tickets… Got him twice plus Mordred, Atalante and Astolfo.

So yeah, don’t believe in summon simulator.

Back to the original question

Well, this would depend on your definition of “usually”. We can break this down statistically for the odds of pulling any gold servant, period, by adding up the odds for every possible scenario:

(odds of pulling 4* and no 5*) + (odds of pulling 5* and no 4*) + (odds of pulling both 4* and 5*)

Using the formulas we have above, that would be something like
100*((1-.97^x)(.99^x) + (1-.99^x)(.97^x) + (1-.97^x)(1-.99^x))

Now you can use an online graphing calculator like https://www.desmos.com/calculator to show you the curve (zoom out as necessary) and click on and follow the curve, with x as your # of rolls and y as your percentage to pull a gold servant.

You can see at 18 rolls (54 quartz), you have a 51.77% chance to pull any gold servant, at 33 rolls (99 quartz) you have a 73.73% chance. You don’t hit a 90% chance until 57 rolls (171 quartz). You could theoretically roll 114 times (342 quartz) and be in that unlucky 1% which still fails to hit a gold servant.

Probability curve should look something like this:

Summary

Mathematicians are free to correct me, my statistics knowledge is pretty rusty.