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This one's probably going to be an odd question and I don't know if someone's already done this, but thought I might throw this out as a question to any math/probability nerds around (and all glory and honor to thee, math nerds, because even though I'm a university history major most high school math lies far beyond my grasp) but I was trying to work this out for a friend of mine who's been stockpiling her orbs.  She's waiting on some of her faves (one blue and one green both in the 4*-5* group specifically) and I'm trying to figure out if it works out better from a probability point of view for her to just stick to summoning blue and green or if it works out better from a probability vs. cost point of view to just keep doing a full summon each time.

So, trying to work it out myself, I started with identifying the number of each unit in-game by color and star:

5* - Red 18, Blue 12, Green 9, Colorless 9

4* - Red 26, Blue 19, Green 15, Colorless 20

3* - Red 15, Blue 12, Green 9, Colorless 13

And then I tried to break each down to a percentage chance per unit by color:

Red Orb:

3% for Focus (if applicable)

3% for 5*, 1/18 of 3% which would basically work out to 0.167% per character

36% for 4*, 1/26 of 36% which would basically work out to 1.38% per character

61% for 3*, 1/15 of 61% which would basically work out to 4.07% per character

Blue Orb:

3% for Focus (if applicable)

3% for 5*, 1/12 of 3% which would basically work out to 0.25% per character

36% for 4*, 1/19 of 36% which would basically work out to 1.89% per character

61% for 3*, 1/12 of 61% which would basically work out to 5.08% per character

Green Orb:

3% for Focus (if applicable)

3% for 5*, 1/9 of 3% which would basically work out to 0.33% per character

36% for 4*, 1/15 of 36% which would basically work out to 2.4% per character

61% for 3*, 1/9 of 61% which would basically work out to 6.78% per character

Colorless:

3% for Focus (if applicable)

3% for 5*, 1/9 of 3% which would basically work out to 0.33% per character

36% for 4*, 1/20 of 36% which would basically work out to 1.8% per character

61% for 3*, 1/13 of 61% which would basically work out to 4.69% per character

But this leaves me with a few questions that I was hoping anyone who's been working with stats for the game so far would be able to clarify: If the color isn't in focus, does that 3% just not apply?  Is there an equal chance of getting the four different colors in a five-summon group or is it distributed based on the number of units in each category? (With higher chance for example of red showing up in a summoning block than, say, green?)  And if one could work out an overage or a percentage of the number of times that a color would appear in the summoning group, could one then use this to more or less predict, if dumping a set number of orbs in (say 100 orbs in one batch for example), what the probability would be of coming up with certain characters?

Most of what I'm managing to accomplish with this is hurting my brain at this point, lol, but I'm wondering if anyone could either fill in some blanks or make corrections to where I'm making any wrong assumptions or guesswork on this.

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No proof, but it's almost certain that the game simply picks 5 characters from the full pool, drops them on the table, and puts big, colorful circles on top of them to hide who they are (or any other method that results in the same resulting probabilities).

Why is it almost certain? Because of Occam's Razor: It's the simplest solution that accounts for things like "what if there is no focus of a given color?" while also making the overall probabilities (assuming the player pulls all 5 characters from each set) exactly as advertised (which has legal implications in some countries).

For Blazing Shadows, you'd end up with something like this:

P(focus) = 3%, n = 6, P(any specific focus) = 3% / 6 = 0.5%
P(5-star) = 3%, n = 48, P(any specific 5-star) = 3% / 48 = 0.0625%
P(4-star) = 36%, n = 80, P(any specific 4-star) = 36% / 80 = 0.45%
P(3-star) = 58%, n = 49, P (any specific 3-star) = 58% / 49 = 1.18%

P(any one orb is red) = 0.5% + 0.0625% x 18 + 0.45% x 26 + 1.18% x 15 = 31.1%
P(any one orb is blue) = 0.5% + 0.0625% x 12 + 0.45% x 19 + 1.18% x 12 = 24.0%
P(any one orb is green) = 0.0625 x 9 + 0.45% x 15 + 1.18% x 9 = 18.0%
P(any one orb is colorless) = 0.5% x 4 + 0.0625% x 9 + 0.45% x 20 + 1.18% x 13 = 27.0%

P(Karel | red) = P(Karel) / P(red) = 0.5% / 31.1% = 1 / 62.2 = 1.61%
P(Ninian | blue) = P(Ninian) / P(blue) = 0.5% / 24.0% = 1 / 48.0 = 2.08%
P(Jaffar | colorless) = P(Jaffar) / P(colorless) = 0.05% / 27.0% = 1 / 53.9 = 1.86%
P(Jaffar U Rebecca U Lucius U Priscilla | colorless) = 4 x 1.86% = 7.42%

etc.

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8 minutes ago, Ice Dragon said:

 

P(Karel | red) = P(Karel) / P(red) = 0.5% / 31.1% = 1 / 62.2 = 1.61%
P(Ninian | blue) = P(Ninian) / P(blue) = 0.5% / 24.0% = 1 / 48.0 = 2.08%
P(Jaffar | colorless) = P(Jaffar) / P(colorless) = 0.05% / 27.0% = 1 / 53.9 = 1.86%
P(Jaffar U Rebecca U Lucius U Priscilla | colorless) = 4 x 1.86% = 7.42%

etc.

somehow i have managed to pull 9 lucius but only 2 jaffars xD it is kinda sad tbh but hey got to roll with what you get am i right xD

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I agree with the "pull from full pool of a given rarity and just hide them behind circles" theory--it's also the only way the advertised "each character in each tier have the same probability of being pulled as one another" thing is possible while being also very simple to implement.

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3 minutes ago, Shiro said:

somehow i have managed to pull 9 lucius but only 2 jaffars xD it is kinda sad tbh but hey got to roll with what you get am i right xD

When your data set is small, variation due to random occurrence is large.

I mean, I have 4 copies of Lucius, but only 1 each of Rebecca, Priscilla, and Jaffar.

Also, out of all of my Blazing Shadows blue orb pulls so far, I've gotten 6 copies of Ninian (2.08% each), but 4 each of Hinoka and 5-star Nowi (0.260% each).

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1 minute ago, Ice Dragon said:

When your data set is small, variation due to random occurrence is large.

I mean, I have 4 copies of Lucius, but only 1 each of Rebecca, Priscilla, and Jaffar.

Also, out of all of my Blazing Shadows blue orb pulls so far, I've gotten 6 copies of Ninian (2.08% each), but 4 each of Hinoka and 5-star Nowi (0.260% each).

yeah my luck just sucks is all wish it was 9 jaffars not 9 lucius and i only got 3 ninians most likely due to the fact that i was focusing the colorless orbs most.

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The math is broken down pretty well here, so even though your friends isn't trying to get 5* units, this should help out.

Spoiler

 

 

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