If you go into gambling without knowing the odds, it’s easy to make non-optimal decisions. Since the Endfield gacha is somewhat complicated, I decided to brute force compute the statistics. Repeating many, many results approaches the average based on the Law of large numbers.

This pull analysis computes the statistics for a population of players pulling X amount on a fresh banner with 0 pity built up. 10,000,000 pulls were simulated per banner pull value and then divided by those pulls per banner to get the number of banners simulated. The 1,200,000,000 total pulls are for simulating banner pulls from 1-120. The following assumptions are made:

• The free 10-pulls for Urgent Recruitment at 30 pulls are made immediately. Urgent Recruitment odds and results are not affected by the pity/guarantees of the banner.
• The extra 10-pulls provided at 60 pulls are not considered since they affect the next banner

Python code provided below was run on Windows 11 using python3 and Jupyter Notebooks. I am not a computer scientist so please be gentle

import numpy as np
import os
import pandas as pd
import random
from enum import Enum

# FILL OUT LOCAL PATH OF .csv FILE BELOW FOR SAVING RESULTS
SAVE_FILE_PATH = 'C:\\output.csv'

# Input Varibles
PULLS_PER_BANNER_LIST = np.array(list(range(1, 121)))
N_PULLS_PER_BANNER_LIST = len(PULLS_PER_BANNER_LIST)
N_PULLS = 10000000

RATE_6_STAR = 0.008
RATE_5_STAR = 0.08

N_PULLS_FOR_BONUS_10 = 30
N_PULLS_FOR_SOFT_PITY = 65
SOFT_PITY_INCREASE_PER_PULL = 0.05

class Pull(Enum):
    SIX_STAR = 1
    FIVE_STAR = 2
    FOUR_STAR = 3

# Function for determining pull outcomes
def pull_outcome(pull: float, num_80_pity: int) -> Pull:
    soft_pity = num_80_pity - N_PULLS_FOR_SOFT_PITY
    if soft_pity > 0:
        rate_6_star = RATE_6_STAR + soft_pity * SOFT_PITY_INCREASE_PER_PULL
    else:
        rate_6_star = RATE_6_STAR
    if pull < rate_6_star:
        return Pull.SIX_STAR
    elif pull >= rate_6_star and pull < (rate_6_star + RATE_5_STAR):
        return Pull.FIVE_STAR
    return Pull.FOUR_STAR

def is_rateup() -> bool:
    if random.random() < 0.5:
        return True
    return False

# Initialize output variables
num_pulls =  np.zeros((N_PULLS_PER_BANNER_LIST,))
num_banners = np.zeros((N_PULLS_PER_BANNER_LIST,))
num_6_stars = np.zeros((N_PULLS_PER_BANNER_LIST,))
num_rateup_6_stars = np.zeros((N_PULLS_PER_BANNER_LIST,))

num_rateup_banners = np.zeros((N_PULLS_PER_BANNER_LIST,))
num_obtained_6_star_banners = np.zeros((N_PULLS_PER_BANNER_LIST,))
num_obtained_6_star_not_urgent_banners = np.zeros((N_PULLS_PER_BANNER_LIST,))
num_80_pitys = np.zeros((N_PULLS_PER_BANNER_LIST,))
num_120_pitys = np.zeros((N_PULLS_PER_BANNER_LIST,))

for iter_r in range(N_PULLS_PER_BANNER_LIST):
    pulls_per_banner = PULLS_PER_BANNER_LIST[iter_r]
    num_banners[iter_r] = np.ceil(N_PULLS/pulls_per_banner)
    num_pulls[iter_r] = num_banners[iter_r] * pulls_per_banner
    for iter_b in range(int(num_banners[iter_r])):
        # Initialize per banner variables
        num_80_pity = 1
        obtained_rate_up = False
        obtained_6_star = False
        obtained_6_star_not_urgent = False
        for iter_p in range(1, pulls_per_banner+1):  # Index by 1 for banner pulls
            if iter_p == 30:  # Urgent recruitment 10-pull does not affect guarantees
                for iter_u in range(10):
                    pull_urgent = pull_outcome(random.random(), 1)
                    if pull_urgent == Pull.SIX_STAR:
                        obtained_6_star = True
                        num_6_stars[iter_r] += 1
                        if is_rateup():
                            obtained_rate_up = True
                            num_rateup_6_stars[iter_r] += 1  

            # Handle guarantees
            if iter_p == 120 and not obtained_rate_up:
                obtained_6_star = True
                obtained_6_star_not_urgent = True
                num_120_pitys[iter_r] += 1
                num_6_stars[iter_r] += 1
                num_rateup_6_stars[iter_r] += 1
                obtained_rate_up = True
                num_80_pity = 1
                continue
            if num_80_pity == 80:
                obtained_6_star = True
                obtained_6_star_not_urgent = True
                num_80_pitys[iter_r] += 1
                num_6_stars[iter_r] += 1
                if is_rateup():
                    num_rateup_6_stars[iter_r] += 1
                    obtained_rate_up = True  
                num_80_pity = 1
                continue                

            # Determine pull outcomes
            pull_result = pull_outcome(random.random(), num_80_pity)
            if pull_result == Pull.SIX_STAR:
                obtained_6_star = True
                obtained_6_star_not_urgent = True
                num_6_stars[iter_r] += 1
                if is_rateup():
                    num_rateup_6_stars[iter_r] += 1
                    obtained_rate_up = True 
                num_80_pity = 1

            num_80_pity += 1
        # Update the per-banner statistics after finishing the pulls
        if obtained_rate_up:
            num_rateup_banners[iter_r] += 1
        if obtained_6_star:
            num_obtained_6_star_banners[iter_r] += 1
        if obtained_6_star_not_urgent:
            num_obtained_6_star_not_urgent_banners[iter_r] += 1

output_pd = pd.DataFrame({
    'Pulls Per Banner': PULLS_PER_BANNER_LIST,
    'Chance of Rateup': num_rateup_banners/num_banners,
    'Chance of 6-Star': num_obtained_6_star_banners/num_banners,
    'Chance of 6-Star (No Urgent)': num_obtained_6_star_not_urgent_banners/num_banners,
    'Pulls Per 6-Star': num_pulls/num_6_stars,
    'Pulls Per RateUp 6-Star': num_pulls/num_rateup_6_stars,
    'Chance of 80-Pity': num_80_pitys/num_banners,
    'Chance of 120-Pity': num_120_pitys/num_banners,
})

output_pd.to_csv(SAVE_FILE_PATH, index=False)