Fes Setsuna: how difficult is it to get her in the gacha?

I have been playing Love Live School Idol Festival All Stars (SIFAS) for the past couple of months. SIFAS is a mobile rhythm game where players build teams composed of cards and tap notes during songs to get high scores. These cards have different skills, attributes and stats and come in different rarities. R (rare) cards are common but weak while SR (super rare) and UR (ultra rare) cards are rarer but more powerful. These cards can be picked up (or pulled) by gacha, where a player spends Star Gems, the in-game currency, to receive random cards. Rate-up cards are new cards added to the gacha that initially receive a slightly increased chance of pulling them for a limited period of time.

SIFAS players have been excited by one particular UR card that was released recently. That card is Fes Setsuna, a very powerful card with excellent stats and great skills. When the card was added in late May, many players (myself included) tried to pull for it in the gacha. While some players walked away with one or more copies of Fes Setsuna, others have not pulled that card, leaving them disappointed.

In this blog post, I will explain the chances of getting Fes Setsuna (or a specific rate-up UR card) and highlight the different possibilities that can occur after pulling a certain number of cards. Along the way, I will explain binomial probability and how it can be applied to this situation. This blog post will appeal to both SIFAS players that want to console themselves on how rare it is to get Fes Setsuna and non-SIFAS players that want to know how probabilities can be applied in real-life.

Introducing binomial probability

There are two possible events in flipping a coin: heads or tails, both with a 50% chance of appearing. If someone flips a coin and we say heads, there is a 50% chance that we will be correct (getting a heads) and a 50% chance that we will be incorrect (getting tails instead). We can flip the same coin twice and calculate the probability of getting a certain number of heads or tails (see the above picture). For example, the probability of getting two heads in two coin flips is ¼. This means there is a ¾ chance of failure (either getting 2 tails or 1 heads and 1 tails).

Binomial probability calculates the chances of getting a certain number of successful events out of a total number of events when there are only two possibilities: success or failure. In the case of Fes Setsuna (or a specific rate-up UR card), there are two possibilities in pulling a card: either we get a Fes Setsuna (success) or we do not (failure). In terms of probabilities, there is a 0.5% chance that we get a Fes Setsuna card (or a specific rate-up UR card) and a 99.5% chance that we do not. We can input these values in binomial probability to calculate the chances of getting a particular number of Fes Setsuna cards out of pulling many cards.

Did you know? For this scenario, we can calculate binomial probabilities via the formula:

where N is the total number of cards pulled and F is the number of copies of a specific rate-up UR card we want to obtain (in this case, Fes Setsuna). 0.005 is the probability of success (get a Fes Setsuna) and 0.995 is the probability of failure (do not get a Fes Setsuna).

The chances of getting Fes Setsuna out of 10 cards

Let’s assume that we pull 10 cards in one go. What are the chances of getting a certain number of Fes Setsuna cards?

Let’s first calculate the probability that we do not pull any Fes Setsuna cards. To do this, we multiply 0.995 (the failure probability) by itself 10 times () to get 0.9511. This means there is approximately a 95% chance that we will not get Fes Setsuna in a pull of 10 cards.

The chances of getting a Fes Setsuna card drop off massively from here. For example, there is a 4.78% chance that you will only get 1 Fes Setsuna card out of 10 (this is calculated by ). And if we were to get 2 Fes Setsuna cards, the chances are even lower at 0.11% (). Hence, anyone getting two or more Fes Setsuna cards out of 10 cards is “very lucky” indeed. 

The probability that we get at least one (i.e., one or more) Fes Setsuna card out of N cards can be calculated by the equation . Essentially, we subtract 1 by the probability of not getting any Fes Setsuna cards in N cards. If we pull 10 cards, then the probability of getting at least one Fes Setsuna card is . This means there is a 4.89% chance that you will get at least one Fes Setsuna card out of 10 cards. These are not very good odds for getting Fes Setsuna.

What if I pull more cards? How likely will I get Fes Setsuna?

We can apply the same formula to calculate the probability of getting at least one Fes Setsuna card out of N cards. I have graphed the probabilities of getting at least a certain number of rate-up UR cards (e.g., Fes Setsuna), up to 250 cards. This is because at 250 cards, the player can “buy” or “spark” the rate-up UR card, essentially skipping the gacha altogether.

As shown in the graph above, as one pulls more cards in a single gacha, the chances of getting a Fes Setsuna (or a specific rate-up UR) card steadily increase. This comes to the point where pulling 140 cards gives a 50% chance of pulling at least one Fes Setsuna card while pulling 250 cards gives a 71.44% chance. Nevertheless, even at 250 cards (the point at which you can “buy” Fes Setsuna), there is still a high chance (28.56% to be precise) that a player will not get a Fes Setsuna card from the gacha.

The chances of getting more than one Fes Setsuna card decrease as we further restrict the possible successful events. For instance, the chances of getting at least two Fes Setsuna cards out of 250 cards is halved from 71.43% to 35.56%. What this graph shows is that pulling more cards in a gacha increases the chances of getting Fes Setsuna. However, it does not exclude the possibility of not getting Fes Setsuna despite pulling so many cards. This is something to be mindful of when spending stars to pull a particular rate-up UR card in a gacha.

Getting Fes Setsuna through simulations

What if you are still not convinced of the possibilities that belie a gacha? We can run simulations to visualize whether and how often possible events occur when they are repeated. In this case, we can simulate how many players pull a certain number of Fes Setsuna cards.

Let’s assume that we ask 100 people to pull either 50, 150 or 250 cards in SIFAS and record the number of people who get one or more Fes Setsuna cards. We can simulate this scenario by using the following code in R.

# Set seed number (so that results can be replicated in someone else's computer)
set.seed(124)

# Generate a table that simulates 100 times the number of rate up UR cards 
# that would be pulled out of 50, 150 and 250 cards. 
trials <- tibble(trial = 1:100, 
                 num_rate_sm = rbinom(n = 100, size = 50, prob = 0.005),
                 num_rate_med = rbinom(n = 100, size = 150, prob = 0.005),
                 num_rate_max = rbinom(n = 100, size = 250, prob = 0.005))

We can see from the graph above that if we only pulled 50 cards, around 80 people will not get a Fes Setsuna card while 20 will get one. This is in line with the 77.83% chance that no Fes Setsuna card will be pulled out of 50 cards. As the total number of cards pulled increases, the number of people getting at least one Fes Setsuna card increase. For example, if we pulled 150 cards where we predict an approximately 50% chance of getting at least one Fes Setsuna, 46 people will not get Fes Setsuna while 54 will. And if we increase the total number of cards to 250 cards where there is a 28.56% chance of Fes Setsuna not appearing, 29 people will not have a Fes Setsuna card through the gacha.

These simulations show the range of possibilities that can occur when pulling cards in a gacha. While there will be some people that will get one or more Fes Setsuna cards, others will not pull any. At no point is it 100% certain that you will eventually get Fes Setsuna through the gacha.

Conclusion

Getting a Fes Setsuna card (or any rate-up UR card) is hard and there is always the chance that you will not get the card from the gacha, regardless of how many cards you pulled. Given that one needs to pull a lot of cards before they can “buy” Fes Setsuna, most people will feel disappointed that they did not get her during the gacha. However, it is not the end of the world if you don’t get that card. What is important is working with whatever cards one has to build great teams that can produce high scores. I hope that this blog post gives you some insights on how difficult it is to get Fes Setsuna in the first place and how binomial probability can be applied to this scenario.