Assessment of Odds: NP-Five-ing a Limited Four-Star in One Go

I plan to roll for Summer Wu next year.

I plan to roll with great intensity for Summer Wu next year.

I actually would like to take her to NP-5 in one session if at all possible, because unlike OG Wu, she’s not going to be able to spook herself* to NP-6 as OG Wu has done for me as of this month.


Sort of. NP-1 was acquired through direct rolling. NP-2 and 3 were from spooks, NP-4 and 5 were acquired from direct rolling, NP-6 was another spook. So literally half-and-half.

So I’m going to throw out a classic question to the number crunchers out there.

What’s the general consensus for how much Quartz it takes to get a rate-up 4* Limited-Time Servant straight up to NP-5 in one session, with, I’d estimate, B+ Luck?


On solo rate up around 1,050. On dual rate up around 1,250. This assumes statistical average on pulls.


I just did some math in my head and based what it took to get my copies of Osakarcher, CARmilla, Jane, Brynzerker, and Saito, you’re looking at a minimum of 12 multis to hit at least NP6.

You could get very lucky and hit a few double SR rolls along the way, but I think you’ll be lucky to hit an NP5 Limited SR in less than 360 SQ (I’ve done it twice). I did not get the SSR until much later, if at all (Himiko did not arrive).

It’s probably closer to 800, which is like 4.5 $80 packs.

Ideally, you’ll get the SSR at least once along the way. Fortunately, Summer Skadi and the 5-star CE are both excellent.


Statistical pull is key. Unfortunately GP is host to a master who got np4 summer Abby and np2 summer tomoe in 1300 sq. On a solo 4* rate up.

Tomoe was the target they saved up months and months for. Aaaand… They hate Abby with a passion. Even more now.


A 4* on single rate up appears in 1.5% of all rolls on average. You will likely get one making approximately 1/0.015 = 66.7 rolls. If you want 5 you will most likely need approximately 333.3 rolls, which at at 30 SQ apiece for 11 rolls yields about 900 SQ required.

That is for roughly a fifty percent chance of getting NP5. Presumably you want better odds than that so you’ll need to save past 900SQ to have more confidence of hitting your target.


Really large amount of Quartz…

Also, Summer Wu will be the only Rate-up 4* on her Banner, and she’ll share it with 5* Summer Skadi.

Really low amount of Quartz…

Really realistic amount of Quartz…

Hahaha, yeah… right. Scathach hates me. Every single attempt I’ve made for her OG and Caster forms have snubbed me something fierce. I expect this session to go the way yours did with Himiko.

Super Oof.

Something about Summer Abby… I managed to get her as well, without even trying. But unlike Scathach, Abby adores me. Probably because we’re actually the most compatible Master-Servant pairing of all the Servants, if that poll I took was any indication.

Also a prime example of the wonkiness of the rates here.

Harsh realities… But also accurate.


Between Melusine and Summer Wu, there are only three rather minor targets on my list, and I’ll most likely only use Tickets for them. Since I’m a Manatee, and depending on how much damage it takes to at least get NP-2 Melusine, I should have… Oh, maybe, 3000 Quartz by the time Summer Wu arrives? Not completely sure, that may be a lot higher than what will be correct, since I haven’t actually run the numbers yet. That should probably be what I should do next.

I will stop at nothing to get NP-5 Summer Wu… …well, within the actual limits, anyway.

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The difference between Infection’s results and my own is that Inflection’s bet on the 50% being in your favor meaning you get hit/spook rather than spook/hit. Mine assumes you’ll get spooks first so requires more investment, it also assumes the 3% chance will always come at the tail end of when it’s supposed to.

But yes, an NP5 SR is a shit show on top of FGO’s larger rate issues also being a shitshow. I did this math months ago because I’m aiming for NP5 Barghest so take that for what it’s worth. The true answer is you can never really save enough to guarantee anything, so save what you can but don’t dodge a banner that might hurt because there’s no guarantee that the extra saving will amount to anything.


The general mantra for FGO Summoning, really.

I wish you the best of success in getting Barghest.

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Another great place to look for when asking yourself probability of [result] questions is this thread by Lexi. I still use it to this day. It just has useful Wolfram links to calculate the chances, specially useful is the one for the chance of [n] amount of the unit for [X] amount of rolls.

So like for 330 rolls (900SQ) at 1.5%:

The chances of at least [n] NP level are given. For NP5 (the rightmost number) that’s little above 50% for 900SQ which is around what Inflection said above.

Note: Definitely would recommend saving more though. We all know from first-hand experience that the actual result can vary very far from the expected so best safe than sorry. 1.2K SQ would give ~80% chance of NP5 which is more reasonable. Others suggested something around this range for confidence too.


I see big equations and numbers like that, and my eyes start to turn into swirls. …normally. Somehow, I actually kind of knew what I was looking at with that, so that thing must be super user-friendly.

Thank goodness there are no major targets between Melusine and Summer Wu, then.

A saving I will go, a saving I will go…

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I didn’t assume anything of the sort. I gave the threshold at which the expected number of rate-up 4* Servants summoned is 5.

I do this for a living, Aardvark.


Just now, you sound just like Holmes. I can even see you doing his signature eyebrow arch.


I’ve re-scanned the list of Servants between Melusine and Summer Wu.

Excluding Welfares, a certain FP Servant, and a certain [][][][][], there are 35 Servants between them.

Of those 35, only the Trung Sisters (cause Pokemon Trainers in FGO is hilarious) and Daikokuten (cute mouse girls!) intrigue me, the latter a bit moreso.

I’ve checked both versions of Chihuahua, and… yeah, I’m not seeing that working out, despite how much luck I tend to have with the Tamamos (NP-3 Mae, IIRC; NP-6 Cat). Maybe some super tertiary attempts, but otherwise, easy enough to move beyond.

It’s going to be a long road to Summer Wu…

And here I’m talking about this before Melusine even arrives. Time to get back to the current year.

Thanks for all the replies, everyone. Here’s to hoping things go well for us all in 2023.


300 rolls at 3% is 9 results which could statistically average out to NP5, but could just as easily average to NP4. Need 333 to average 10 and thus NP5. I assume bad luck and further guess that the first result isn’t statistically likely because odds from a previous session might delay the expected first result until roll 66 and thus the last at roll 366 as a cushion.

That’s what I’m talking about.

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The expected number of successes in a Bernoulli trial is not a matter that varies from one situation to another. There is a mean, and there is a variance. Also, I have no idea why you are using a figure of 3% or what you mean by 9 results in this context.

Need 333 to average 10 and thus NP5. I assume bad luck and further guess that the first result isn’t statistically likely because odds from a previous session might delay the expected first result until roll 66 and thus the last at roll 366 as a cushion.

That’s what I’m talking about.

Then you are, as far as I can understand your reasoning, giving numbers that seek a high confidence of achieving the desired target. I specified in my post that the number I gave yielded a 50% chance of NP5.

Here are the exact numbers. You can plug that formula into wolframalpha to get this graph yourself. At 1200 SQ, yielding 440 rolls, the chance of getting a 4* Servant on 1.5% rate-up to NP5 is 79.6%.


3% is SR rate overall. 1.5% is the solo rate up chance. Simpler to explain it to people as a 50% chance any SR you get will be the rate up.

No just explaining the average is a range. Low end is 9 results, high end assumes you stopped rolling on a high note(decently likely) and thus are due a spell of nothing good for a while basically stating that average is that 9-11 SR rolls at roughly 33 rolls per SR. So 300-366 rolls depending on what you ended your last roll session on and whether the odd-numbered coin flip favors you or not.

There is no such thing as confidence in rolling in this game, only reasonable saving. I err on the side of the worst average range and save for that.

Absolute average would be 333(10 results at 50% chance of getting what you want average) I guess which would be what 999 SQ given 3 SQ is 1 roll?

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This is the gambler’s fallacy.


How? If in 33 rolls you’re likely to get 1 good result and you stop on that good result there’s a chance(slim granted) that your next 32 will be bad. Odds average out over time and the bad results have to show up sometime.

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Because future rolls are not determined by previous rolls. If between two gamblers expending the same number of resources on a game we are told that one had good luck in the first part of the session, then that gambler is likely to do better overall because both gamblers will do statistically equally well in the remaining portion of the game, and we have information that one did better early.

You are also collecting rolls into groups and treating them as discrete groups in which the statistical behavior is guaranteed: “If the good roll didn’t show up early, it will show up late.” That isn’t how it works. Even if you are committing the gambler’s fallacy conservatively against the gambler, you are still committing it.


In the larger scheme yes there’s no guaranteed results, but we’re averaging results. I can’t know the person’s average roll history so I have no idea which end of the spectrum they’re currently swinging towards. So I assume mostly neutral with the tendency to roll until you get what you want. Statistics average out over time, so again assuming a more conservative average. Not saying that’s a guarantee, I’m sharing how I save and I came up with these numbers because when I assume the negative end of statistical neutral it bears results more frequently than assuming pure neutral.

Yes that’s stacking odds and yes it will slightly tip the scales, but psychology is also a part of gacha rolling and banking on the worse end of average is better for my sanity personally. It allows me to have a goal that’s not betting the farm because there’s no guarantees with SRs, I could roll 10,000 times and get nothing. If I did that’s a deal breaker for gameplay because it’s extremely disheartening. It’s also disheartening to not get what you roll for. So compromise, save for the average then assume I’m going into the average with a karmic debt if you will. If I still fail then thems the breaks I did what I could within reason and erring the way I do makes it feel as if I did legitimately try so there was nothing I really could do to make it better.

So yes I suppose a bit of mental gymnastics are involved, but at the point where the difference in our answers is basically 30 rolls it doesn’t matter that much. The answers boil down to “save 333 rolls for average” vs. “save 366 rolls so you won’t torture yourself over just 1 or 2 more multis” which I think are both fair takes. Just depends on the mentality.

I always assume I’m due for a bit of bad luck because being pleasantly surprised is a better feeling.

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Okay, I’ll conclude my participation in this part of the conversation with this point. No one is ever swinging towards either end of the pendulum. To persist in reasoning using this intuition will always be false numerically.

But ultimately, that’s not your final measure. It sounds to me like you basically begin with the usual statistical calculations and then sandbag your expectations, and this is how you spend an amount that’s satisfactory to you. I’m not going to argue with however someone wants to have fun. May you frequently be pleasantly surprised.