FEH Summoner: How lucky are you?

Since I got tired of the Gacha (translate: got salty) in Fire Emblem Heroes, I thought I’d spend a bit of my time to investigate the probability and the statistics of my luck on Gacha itself.

There are some summon simulator tool out there that tells us how many orbs we need to summon any 5 star heroes, focus heroes, etc. It even tells us the average orbs needed, median, 90th percentile, etc.

However, none of those tools can tell me where my gacha luck fall. Do I have an average, above average or below average luck in Fire Emblem Heroes gacha. So, I wrote myself a program to simulate the outcome of my typical summoning session. Then import the data to excel and chart the result. For simplicity, I use the following scenario:

– Pull every single stone in the circle (no sniping)
– Spend 20, 40, 60, 80, 100, 120, 200, 300 and 600 orbs
– Summon until orbs are gone in 1 session. No break
– No color preference. Random 5-star or any focus heroes will get counted
– Use a sample size of 1,000 people
– Simulate regular banner (3% focus, 3% 5-star)
– Simulate legendary banner (8% focus, 0% 5-star)

I didn’t record any of my past summoning session result. So, I can only make a best guess (estimate) on how I typically did the summon. As far as I can remember, I mostly summon whenever I have about 20 orbs. Sometimes 40 orbs. Then, after I learned about the average number of orbs needed for 50% chance of getting a 5-star, I started to spend 60 or 80 orbs on many banners. I spend 120-140 orbs a few times. I spent about 200 orbs once on a legendary banner. I summon mostly on new banners and legendary banner. I don’t typically summon on re-run banner.

Regular Banner

Let’s see the simulation result and the luck distribution among 1,000 people.

This is the typical outcome for 1,000 people who spent 20 orbs worth on a banner (full circle, or 16 orb if we discount the first free summon).

Note on how to read the chart:

– X-axis shows the number of 5 stars we get from the summoning session
– Y-axis shows the number of people
– FocusCount on the chart means that chart shows the distribution of people who get random focus heroes
– PityCount on the chart means that chart shows the distribution of people who get random pity broken heroes
– AnyCount on the chart means that chart shows the distribution of people who get any random 5-star heroes (focus and pity broken heroes are grouped together)
– Basically, on a standard bell curve, less luck people will be on the left side of the chart. While luckier people will be on the right side of the chart. The highest column represents the average luck people.

As you can see from the chart, about 75% of the people will not get any 5-star heroes (0 focus and 0 pity broken heroes). About 86% of the people will not get any of the 5-star focus heroes. Only 13% of the people get one 5-star focus heroes and less than 1% of the people got super lucky with two 5-star focus heroes. About 12% people got pity broken. Less than 1% got 2 off-focus 5 star heroes.

Here’s the typical result if we spend 40 orbs on a banner.

There are more people that get either a focus or pity broken heroes. About 25% of the people get either a focus heroes or a pity broken heroes.

As we increase the number of orbs spent, we can see the graph morph to become a standard bell curve. Graphs below show the luck distribution when we spend 60, 80, 120, 200, 300 and 600 orbs. The bell curve started to shows at about 200 orbs spent. At 600, the curve looks like a standard bell curve.

So, how did my summoning session on a regular banner do in the past? Where does my luck fall? I can’t really tell as I didn’t spend enough orb to know where I land. With the way I summon in the past, the graph is skewed more to the left. The maximum amount of orb I spent on a regular banner was 120-140 orbs. Based on the chart, there are more than 33% of people who don’t get any focus heroes. About 34% of people get 1 focus heroes and about 22% get 2 focus heroes. I didn’t get any focus heroes when I spent 120-140 orbs. I got pity broken. I’d say my luck is not that good.

Legendary Banner

How does the distribution look like on legendary banner?

Since the rates in legendary banner is a little bit better than regular banner, we can see the bell curve to form starting at 80 orbs and it becomes smooth at about 120 orbs.

On most legendary banner, I spent either 60 or 80 orbs. Most of my session ended with me getting either 1 focus heroes or nothing. In the last couple months, I spent 200 orbs and 120 orbs on the legendary banner. I got 6 focus heroes and 1 focus hero. So, on April Legendary banner, my luck is slightly above average. While on May, my luck is below average.

Conclusion

So, after doing the math and statistics, here’s what I think:

– If you only have few amount of orbs, don’t summon. I know that we may get lucky, but the chance of not getting any 5-star heroes is too high. You’ll just waste orbs. Just save the orbs. We get about 200-300 F2P orbs per month and banners regularly come back. You don’t need to rush to summon. Just wait a bit until your orb count is high enough.
– If you’re summoning on regular banner, I’d suggest that you’ll need to have at least 300 orbs just to be safe side. People with average luck can expect between 1-4 focus heroes (2-8 heroes if we include pity broken heroes). However, even at 300 orbs, there are still about 3% of people who won’t get any focus heroes. So, prepare to get salty if you’re extremely unlucky.
– It’s better to summon on legendary or mythic banner. We have a higher chance on getting focus heroes. I recommend to only summon on legendary banner if you have more than 200 orbs. Average luck people will get between 2-5 focus heroes. There’s still a 1% chance of not getting any focus heroes. But the chance if very low. Unless you’re extremely unlucky, you’ll get at least 1 focus heroes.
– If you’re extremely lucky (who can get 7 focus heroes in 120 orbs on regular banner), just disregard my findings here.

I only have 32 orbs as of now. I need to take on my own recommendation and force myself to save the orbs until I have enough to summon.

What’s your luck?

So, fellow Fire Emblem Heroes summoner, how did you do in your summoning session? Are you lucky, average or unlucky? Compare with the baseline chart above.

Here’s a link to a few summoning session that other people did in youtube to see how lucky or unlucky they are:

https://www.youtube.com/watch?v=VCSJ91ePfeg
Regular banner – 350+ orbs: 2 focus, 4 pity broken (average luck)

https://www.youtube.com/watch?v=p8RQDXuoHMk
Regular banner – 270+ orbs: 0 focus, 2 pity broken (Extremely unlucky)

https://www.youtube.com/watch?v=jNcUrNNKKe8
Regular banner – 500 orbs: 3 focus, 5 pity broken (below average luck)

https://www.youtube.com/watch?v=n4OIDphc6G4
Legendary banner – 300 orbs: 5 focus (average luck)

https://www.youtube.com/watch?v=hMtqFOGeLWY
Legendary banner – 400+ orbs: 10 focus (above average luck)

https://www.youtube.com/watch?v=iyknfYY3554
Regular banner – 250+ orbs: 1 focus, 2 pity broken (below average luck)

https://www.youtube.com/watch?v=3-geuXyRB68
Regular banner – 300+ orbs: 4 focus, 2 pity broken (above average luck)

https://www.youtube.com/watch?v=KJ4mQ8LCXKc
Regular banner – 400+ orbs: 3 focus, 4 pity broken (average luck)

https://www.youtube.com/watch?v=I1KVx3MGjYs
Legendary banner – 290 orbs: 6 focus (average luck)

https://www.youtube.com/watch?v=IG9Pb-KJ6SI
Regular banner – 170+ orbs: 4 focus, 1 pity broken (Extremely lucky)