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Correction on my math of full-round-and-run

This occurred to me a few days ago but I didn't have the energy to write about it. I'm doing slightly better today and I want to set something straight.

A few weeks ago I posted explaining that finishing the round on a legendary banner is worth it if you like 8/12 focus units, by which I meant 2/3 on every colour. If you like 3/3 on a colour you should pull that colour! But if you only like 2/3 on each colour, I demonstrated that you can finish the round and you'll get a focus unit on average every third banner, with 2/3 focus units being ones you want (again, on average), so 2 wanted focus units in 9 banners, 9 x 15 orbs = 135, 63 orbs a focus unit is pretty reasonable. And I wouldn't say I missed entirely, because mathematically speaking that is indeed the expected result.
https://fireemblem.gamepress.gg/q-a/just-figured-out-math-full-round-and-run-legendary-banners

However, I think it's better not to regard finishing the circle as 5 pulls for 15 orbs but 4 pulls for 15. The free summon is always free on every banner. The 15 orbs only gets you 4 extra pulls. Compared to the previous situation, one of those focus units might have been a free pull, which shouldn't count as part of the 15 orbs.

So what exactly are the probabilities on the 15-for-4? If you like 8/12 focus units, the probability that none of them is a focus unit is 94% ˆ 4 = 78.1%, which is slightly lower than 4/5. So it's a focus unit every 5th banner, which means every 75 orbs. That's still a decent price, but consider that you're not going to be interested in every third unit, and now it's 2 desired focus units for 225 orbs, which is in the neighbourhood of the price for pulling a focus unit from a 3%/3% banner with three focus units and no shared colour (except on a 3%/3% banner you might get a random off-focus to go with your target unit).

What this means is that finishing the round after a free summon isn't worth it even if you like 2/3 focus units on all colours, and if this is true for legendary banners, then it's not going to be a good idea on a 3%/3% banner either, all other things being equal.

Apologies for my inaccuracy the first time around which suggested the exact opposite course of action to what I'm now suggesting. I'm glad I was able to catch it before this legendary banner releases; there was no legendary banner between my initial post and this correction.

Asked by Seeker1 month ago

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Does your corrected math go against the theory that it is likely to get a hero every third 5-for-15 pull?

No; that's what the second paragraph is about, what still stands of my previous analysis. You can still expect a focus unit every third round, but you're not getting above-average value for orbs because it's actually focus unit every 75 orbs, not 45. Free summons should not be included in the pulls you get for 15 orbs.

So the best orb value is still sniping a colour on a 3 hero banner (considering the 2 hero one is an exception for now) or on a Legfest where I like 3/3 heroes?

Sniping on a 3%/3% banner with three focus units isn't best value unless you want both focus 5*s sharing a colour. However, it does give you the lowest cost for an individual unit, so if you're trying to merge some 5* exclusive to +10 that's the cheapest way to do it. (Best value that isn't something very unusual like our current two-focus banner or the last VG banner with all blues would be wait until they appear on a legendary banner with two others that you wouldn't mind getting 10-11 copies of, but when's that going to happen lol)

Best value, barring weird stuff like what I already mentioned, is 3%/3%, three focus units, colour shared, and you want at least 2 of the units sharing colour. Four focus units and you want both units sharing a colour is pretty close to 3/3 on a legendary banner, which is still very good value.