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Know the Odds: A General Probability Analysis of Fire Emblem Heroes

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Introduction

Fire Emblem Heroes (FEH) is a “gacha game”, whereby you spend currency (Orbs) to obtain randomized rewards (Heroes). In most modern gacha games, the rates of obtaining different levels of rewards are public information. The same could be said for FEH. However, FEH does have a slight twist to the gacha mechanic, which has implications for the amount a player would spend. In this article, we will be discussing the odds of obtaining different units from summoning in Fire Emblem Heroes.

Summoning System: What to Know

While there is no confirmation of how the summoning system works in FEH, below is the general consensus of how the units are generated when you press the summon button:

• Units are divided into different pools by star rating, as publicised in the Summoning Rates page.

• Randomly choose a pool based on the given probability.

• From the pool, randomly choose a unit.

• Show the unit’s corresponding color in the summoning circle.

• Repeat step one through four 4 times to obtain the full summoning circle.

The most important detail is the 4th step, whereby the summoning circle will show the colors of the units. This gives additional information to the player, which allows them to have some element of control in their summoning sessions. A red orb will always produce a red unit, so on and so forth. This means that a player has some level of control over the units that they will get.

Let’s say for instance that we take a 3 unit standard focus banner with no color sharing, with the probability of getting a focus unit at 3%. With the above method, if a player has no information on the heroes summoned, they will have an equal chance of obtaining each of the focus (1%). However, because they know the color of the available heroes, the player can choose to only summon in that particular color as much as possible, which eliminates the probability of receiving other focus units from the pool.

This does not mean that the odds of getting a particular unit will be the same base rate. For the above example, even if the player eliminates the other focus units by only rolling for one particular color, it does not mean that the probability of the focus unit will remain at 3%, as the number of available units in each color will be different, which results in a different distribution overall.

The general trend is that red has the most units in the summoning pool, green has the least, and blue and colorless are roughly the same. This means that the odds of not getting a red focus are larger than all the other colors, and the inverse can be said for green focuses. This meant that red focus heroes are harder to pull for, while green focuses are easier.

Another mechanic in Fire Emblem Heroes is the pity system, whereby the base rates will be altered according to the amount of consecutive rolls that a player does not receive a 5-star (I will define this as “pity streak”). Every time the pity streak increases by 5, the 5-star rate will increase by 0.5% that is distributed between focus and non-focus, and conversely, 4-star and 3-star rate will drop by the same proportional amount.

Possible Variables

Now that we have discussed the summoning system, we can now describe the different variables that impact your odds.

Base Rate

The obvious variable that influences your chances is the base rate. The higher the focus rate, the higher your odds will be to obtain a hero in a certain pool. The usual standard banners have a base rate of 3% on focus and non-focus 5-star, 58% on 4-star and 36% on 3-star. There are other possible base rates, such as 8% focus rate and 0% non-focus 5-star for the monthly Legendary Heroes banner.

However, while having a high base rate is definitely a good thing and it raises your chances of getting a unit in the focus pool, it is obviously subjected to other variables as well.

Number of Units in the Focus Pool

As mentioned in the previous section, the base rate only influences the probability of choosing the focus pool itself. After that, there is the second step of generating the particular hero in the pool. Assuming that the probabilities are equally distributed, the number of heroes in the focus pool will have an impact to the unit’s individual probability.

Color Sharing

Color sharing is not exactly a factor that directly increases the expected odds, however, it does indirectly impact the number of orbs spent to obtain a certain character due to the pity system.

If there exists a color sharing in the focus pool, it means that there is an equal chance for a player to get units of the color. Hypothetically, this means that in general, the number of “pity streaks” will definitely drop, which is not a good thing if the player is going for a particular hero.

Introducing Simulations

From the above discussion, we can see the consensus of the summoning system, and the different variables that influence the odds. In the next section, we will be examining the impact of the variables through some basic simulations and the traits of the different types of banners in Fire Emblem Heroes.

We will be conducting two types of simulations. One for getting a single copy of the unit, the other for getting a fully merged copy (essentially 11 copies). If there is color sharing in the banner, we will also simulate for obtaining a single unit for that particular color.

The simulation will assume that we are only summoning a target color (also known as “sniping”). If no target color is found, the simulator will automatically pick a random color and stop summoning. Each trial will complete when the simulation obtains 1 or 11 copies of the unit depending on the simulation.

We will repeat the above simulation for 10,000 trials for single units, and 1,000 trials for full merges. The accuracy could be improved by increasing the number of trials, however, the above numbers give a good approximation of the different variables.

We then sort the simulation results and report the percentile values in units of 5%. This gives a good indication of the probability of getting a certain unit given a certain number of orbs. So, if the 50th-percentile is said to be 100 orbs, it meant that the approximate probability of obtaining the unit in 100 orbs could be said to be 50%.

Simulation 1: Standard Banners

The “standard” banner we will be using is the “Doorway to Destiny” banner. The reason this banner is selected is that not only it shares the same common pool (the standard units that you can pull from any normal banner) with many other banners, but also that it has the simplest case for a banner; 3 units with no color sharing.

We obtain the following results for both single unit and full merges:

 Percentile Red, Solo Blue, Solo Green, Solo Red, Merge Blue, Merge Green, Merge 0% 5 5 5 507 405 404 5% 13 13 10 861 788 702 10% 20 19 18 980 915 795 15% 29 28 24 1057 999 860 20% 38 36 32 1124 1063 921 25% 47 43 39 1191 1128 967 30% 56 52 47 1247 1167 1005 35% 68 61 55 1295 1221 1048 40% 78 71 64 1345 1272 1093 45% 90 82 73 1399 1309 1132 50% 102 93 83 1452 1342 1167 55% 117 106 93 1508 1401 1210 60% 131 121 105 1559 1461 1248 65% 150 137 119 1626 1517 1287 70% 169 156 134 1678 1575 1338 75% 191 176 150 1737 1626 1408 80% 219 202 172 1808 1693 1467 85% 254 237 202 1894 1784 1549 90% 302 286 243 2029 1903 1651 95% 380 367 313 2233 2061 1789 100% 1049 1158 906 3147 2629 2609

From the simulation, we can already make some observations. As mentioned in the previous article, the sheer amount of red units in the game saturated the red pool, resulting in a consistently higher cost for pulling red units throughout all percentages. The opposite could be said about green units.

We also could observe that the variance of rolling for 11 copies as oppose to a single copy is actually smaller. In order to understand what this means, we must take the ratio between the 95th and 50th percentile for comparison. While the amount of orbs for obtaining the full merges is staggering, in relative terms it is actually less likely to be screwed by the random number generator (also known as "RNGesus") as opposed to rolling single copies.

The main takeaway from above is that it is worth it for players to save their orbs and go for full merges due to increased consistency.

Simulation 2: Color Sharing

We now consider the case for color sharing. Consider the Heroes with Fury banner on 6th September 2018, which comprises of Eldigan, Myrrh, and Exalted Chrom. As this banner shares the same common pool as our standard banner, we can use this as a specimen to compare the impact of color sharing specifically for red units. For the sake of this comparison, we will only compile the results of the red units of both the standard banner and the color sharing banner.

The simulation suggests that there is indeed a slight increase in the number of orbs required to reach the desired approximate probability. However, it is also worth noting that while getting the target unit, it is likely that the player might also get the other unit in the banner in the process. Looking at the shared probability, the percentile orb costs are significantly lower compared to targeting an individual unit. At least for individual copies, the increase in orb expenditure can be considered negligible.

However the same could not be said for obtaining full merges. The small number of orbs actually does add up in the case of color sharing. The percentile cost at the lower values does not seem to differ much, however, at higher values, there is a rather significant difference.

 Percentile Solo, Standard Merge, Standard Color Solo, Sharing Merge, Sharing 0% 5 507 5 5 490 5% 13 861 9 13 874 10% 20 980 13 22 1011 15% 29 1057 18 31 1084 20% 38 1124 22 39 1172 25% 47 1191 27 49 1225 30% 56 1247 32 58 1295 35% 68 1295 37 68 1355 40% 78 1345 42 80 1414 45% 90 1399 48 92 1466 50% 102 1452 55 105 1512 55% 117 1508 62 120 1571 60% 131 1559 70 137 1628 65% 150 1626 79 154 1696 70% 169 1678 89 175 1784 75% 191 1737 101 198 1875 80% 219 1808 115 228 1970 85% 254 1894 133 265 2070 90% 302 2029 159 324 2160 95% 380 2233 199 415 2391 100% 1049 3147 648 1150 3238

From the above, it shows that the impact of color sharing is rather minimal for single unit pulls, and it is also compensated by having a relatively high chance of getting the other unit. However, for full merges, a banner with color sharing seems to have a rather significant increase in cost, especially for higher percentile values. It is suggested for players to wait for no color sharing if pulling for merges.

Simulation 3: Four Hero Banners

We now consider the banner for “Arrival of the Brave”, which has 1 unit of each color and shares the same common pool as the standard banner. For simplicity, we will only compare the red unit values.

 Percentile Solo, Standard Merge, Standard Solo, 4-Unit Merge, 4-Unit 0% 5 507 5 675 5% 13 861 15 1141 10% 20 980 27 1272 15% 29 1057 37 1373 20% 38 1124 48 1449 25% 47 1191 61 1510 30% 56 1247 72 1591 35% 68 1295 84 1659 40% 78 1345 98 1744 45% 90 1399 114 1812 50% 102 1452 130 1877 55% 117 1508 147 1953 60% 131 1559 167 2008 65% 150 1626 189 2078 70% 169 1678 213 2160 75% 191 1737 241 2255 80% 219 1808 278 2363 85% 254 1894 326 2495 90% 302 2029 390 2665 95% 380 2233 501 2924 100% 1049 3147 1524 4474

From the simulation, the cost of pulling for the unit has significantly increased for both individual pulls and full merge pulls. This makes a lot of sense as the base rate of 3% is split among 4 units (0.75% for each unit) rather than 3 (1.00% for each unit). What is more interesting is when we compare it to the color sharing values, as the impact of having an additional unit in a banner is much more significant than color sharing.

This suggests that unless you really want a particular character, it is not recommended to pull on 4 unit banners. Fun fact: There used to be a single banner with 6 units, and 4 of them are colorless units. Just let that sink in.

Simulation 4: Hero Fest

The last variable we might want to explore is increasing the base rate. It is unclear whether these kinds of summoning events will be coming back anytime soon, so let’s examine why the developers rarely have these events anymore.

We can emulate a Hero Fest rate by artificially increase the base rate from 3.00% to 5.00% for focus heroes. As Hero Fest tends to be a 4 unit banner, we shall use the units from “Arrival of the Brave” as the data when conducting the simulations.

 Percentile Solo, 4-Unit Merge, 4-Unit Solo, Hero Fest Merge, Hero Fest 0% 5 675 5 460 5% 15 1141 13 733 10% 27 1272 19 806 15% 37 1373 26 871 20% 48 1449 33 925 25% 61 1510 40 967 30% 72 1591 48 1009 35% 84 1659 56 1055 40% 98 1744 65 1110 45% 114 1812 73 1156 50% 130 1877 83 1194 55% 147 1953 94 1232 60% 167 2008 106 1281 65% 189 2078 121 1330 70% 213 2160 137 1376 75% 241 2255 156 1428 80% 278 2363 179 1497 85% 326 2495 208 1586 90% 390 2665 251 1670 95% 501 2924 321 1808 100% 1524 4474 1098 2587

The simulation results provide us with a lot of insight of the rarity of such summoning events. The improvements in obtaining both a single unit and fully merged units are staggering. Hero Fest is pretty much the undisputed champion in terms of individual units and merged copies. Mathematically, the probability of obtaining a particular unit is 1.25%, which is higher than a 3-unit banner’s 1%, which explains the much lower cost.

Simulation 5: Legendary Banners

Let us now mash all the factors together. We will now use Legendary Marth’s debut banner as the dataset. Legendary Banners have a higher base rate of 8% and remove the probability of obtaining a 5-star unit from the standard pool. However, the banner has 12 focus units rather than the usual 3 to 4. As seasonal units usually re-appear in Legendary Banners, and seasonal banners are 4-unit banners, we will use the 4-unit red column as a point of comparison.

We can calculate that the probability of getting a particular unit is roughly 0.66%, and a unit of a particular color at 2%. For reference, 4-unit banner with no color sharing has a probability of 0.75% for each unit. We can predict that the cost of individual units will be much higher for legendary banners, however, the cost of individual colors will be slightly lower.

 Percentile Solo, 4-Unit Merge, 4-Unit Color Solo, Legendary Merge, Legendary 0% 5 507 5 5 686 5% 13 861 9 17 1281 10% 20 980 13 28 1454 15% 29 1057 18 40 1581 20% 38 1124 22 52 1664 25% 47 1191 26 65 1765 30% 56 1247 31 80 1866 35% 68 1295 36 95 1955 40% 78 1345 41 112 2036 45% 90 1399 47 130 2113 50% 102 1452 53 148 2198 55% 117 1508 60 168 2285 60% 131 1559 68 190 2367 65% 150 1626 76 216 2461 70% 169 1678 86 247 2547 75% 191 1737 98 284 2668 80% 219 1808 112 330 2813 85% 254 1894 128 388 2945 90% 302 2029 151 463 3117 95% 380 2233 189 594 3419 100% 1049 3147 535 2247 4682

From the simulation results, the prediction is likely to be correct. It is also worth noting that the quality of Legendary Banners are usually higher than of standard banners in order to entice more players to pull on Legendary Banners. However, in general, Legendary Banners are terrible in terms of accumulating merges. This also meant that merging Legendary Heroes could be extremely costly in the long run.

On FEH Channel April 2019, there were 2 changes that were outlined in the summoning pool.

1. All 5-star unit before Book 2 will be removed from pools with new units. This includes normal and seasonal units.
2. Unit demotions to the 3-star pool for Book 2 units that were in the 4-star pool.

The base rate of obtaining each individual focus remains unchanged, however the change proportion of units in each rarity will indirectly result in a slight change in probability. For this comparison, we will call the pool with units before Book 2 “Legacy” pool, while units that has the units removed the “Standard" Pool.

The calculations below are done after the “Beyond Darkness” banner. Unless the developers speeds up the release of units in the game, it is likely that the proportion below remains relatively consistent.

As of the “Beyond Darkness” banner, below are the proportions of Standard and Legacy Pools.

Pool Proportions. 5☆S - Standard Pool, 5☆L - Legacy Pool. All numbers in %.
5☆S 5☆L 4☆ 3☆
Red 30.4 37.5 28.8 29.4
Blue 28.6 25 27.9 27.5
Green 26.8 22.1 18.0 `7.4
Colorless 14.3 15.4 25.2 25.7

This shift in proportions will result in a change in focus probabilities as well; a larger proportion of units in non-focus pool will result in a lower focus probability. From the above data, we can safely predict that the focus rates of Reds and Colorless units will increase, while Greens and Blues will decrease. The effect is likely to be relatively small however, due to the non-focus rate being much lower than the 4-star and 3-star pool.

We then calculated the exact probability (to 1 decimal point) to obtain the focus units in a hypothetical banner with 1 focus unit pet color. Indeed, the results are congruent to the predictions made above.

Focus Rate after units were generated. All numbers in %.
Standard Legacy
Red 2.590 2.572
Blue 2.706 2.717
Green 4.103 4.135
Colorless 2.993 2.989

The main impact of the removal of units before Book 2 from the standard pool is the quality of units; due to their recency most of the newer units tends to be much stronger and has much better fodder options than their older counterparts. While there is a shift in focus probability, the changes are likely to be rather insignificant.

Conclusion

Factors that influence the base probability of the unit will have a large impact on the expenditure of orbs. This could range from altering the base rate and the number of units in the summoning pool. Color sharing does have an impact, however, it is rather minimal and only could be seen if a player is going for merges. The simulation results also reaffirm the conjecture that red and green units are the hardest and easiest to pull respectively.

Summoning in Fire Emblem Heroes is still based on luck. However, by understanding the mechanics of summoning, we can sometimes shift the odds into our favor, and more importantly, adjust our expectations when we are going for certain units.

May the odds be ever in your favor.