Article
TABLE OF CONTENTS:
0-Introduction: Frustration
1-Binomial Probability Distribution Function
2-Generalized for 1 Equipement
3-q1-q5 Odds for Eastern Ukraine Battle
4-Theoretical BH Calculations
5- Conclusion
0-Introduction: Frustration
Hello sNorway,
If you are reading this article you and are a congressmen in sNorway flipping accept the CS requests. I felt as though I am getting robbed every battle I enter because I have yet to be awarded any item from a post-equipement implemented battle and it's pissing me off. I was wondering if I was just plain unlucky or if the probabilities are weird. This is what I'll be looking into today using the Binomial Probability Distribution (thanks Bernoulli! =P) where success is getting a piece of equipment and failure is not. Needless to say this article contains math, if that's not your cup of tea feel free to leave or skip to the conclusion. All are calculated from my perceptive because I'm selfish.
1- Binomial Probability Distribution Function:
Where:
n = the number of trials
x = number of success(es) for the number of trials (therefore x<=n)
p = probability of success in 1 attempt
q = probability of failure in 1 attempt
Cnx = the number of ways a set can be arranged without order mattering (Cnx = n!/x!(n−x)!)
2- Generalized Formula for 1 Equipement:
I will generalize the formula for our purpose, calculating battle probability assuming I want only 1 piece of equipement for any given quality.
n = the number of attempts you get at an item, this dependes on the battle and is the number of equipement generated
x = 1 because I if I get one piece of equipement I'm flipping happy
p = # of successes / # of attempts
q = 1 - p (the rest of the probability)
Cnx = n!/(n-1)! because x =1
P(X) = n!/(n-1)!*p*q^n-1
3- q1-q5 Odds for Eastern Ukraine Battle
Let me start with the q2 and q1. I hit 60 times in this battle (11 q5 food + day change). I need the total number of hits for the battle this would have been hard to calculate luckily the admins released an article explaining the item generation-- the numbers don't match up. According to admin, there is 1 Q5 item generated for every 30.000 hits, 1 Q4/10.000 hits, 1 Q3/3.000 hits, 1 Q2/1.000 hits, and 1 Q1/300 hits in a given battle however when calculating the total number of hits the battle had I got different results for each item quality. For instance, 53 q1 items means 15900 + (up to 299 more) hits, 16 q2 means 16000 + (up to 999 more) hits, 5 q3 means 15000 + (up to 2999 more), no problems there; we can deduce the number of hits is between 16000 and 16199 however 2 q4 items were generated meaning 20000 hits and 1 q5 item was generated meaning 30000 hits (WTF?!). This is either a glitch in which case I will graciously take my Tester Medal or (more likely) the admins have a probability of getting higher q items with a lower amount of hits and simply did not update how items were generated at the end of a battle.
Anyways, for the purposes of my testing I will suppose there have been 16100 hits total but again remember the number could be anything between 16000 to 16199 as far as I know (it seems unlikely that it is more than 20k). I am furthermore making the assumption that all hits are of equal value (eg admin said hits increased changes of getting an item but not the exact formula they use maybe later rounds hits have more influence for items, for instance).
n = 16 (q2) and 53 (q1) p = 60/16100 = 3/805 (q2/q1) q = 16040/16100 = 802/805
q1: P(X) = 53!/52!*(3/805)*(802/805)^52 = .16266... = 16.27% (roughly)
q2; P(X) = 16!/15!*(3/805)*(802/805)^15 = .05637... = 5.64% (roughly)
To illustrate I will caculate the probabilty I should have gotten a q3 equipement in the following battle: Eastern Ukraine RW where I got into the top 10 a measly 4 times. The total people in the top 10 are 8 rounds * 10 * 2 sides - 4 missed slots = 156 top 10 players (they can be in more than twice). 5 q3 items were offered and remember that the trials are independent of each other and a lucky player can be awarded twice or more for 1 slot.
n = 5, p = 4/156 = 1/39, q = 155/156 = 38/39
P(X) = 5!/4!*(1/39)*(38/39)^4 = .115553... = 11.56% (roughly)
q4 is simpler with me being in the top 3 twice and a total of 8 rounds * top 3 slots * 2 sides = 48 slots. Two q4 items were generated which is weird.
n = 2, p = 2/48 = 1/24, q = 46/48 = 23/24
P(X) = 2!/1!*(1/24)*(23/24)^1 = .07986... = 7.99% (roughly)
q5 is the simplest of all to calculate and does not require the Binomial Probability Formula (due to 1 trial). I did not receive a BH medal but I will calculate as if I did get one. There are 8 rounds * 2 sides = 16 slots.
P(X) = 1/16 = .0625 = 6.25%
4- Theoretical BH Calculations
I will now make a theoretical example where instead of hitting in multiple rounds I would have used my 60 hits to acquire a single BH medal. This will illustrate what the probability of a single chance for the q3, q4, and q5 and give a general idea of the probability.
q1: P(X) = 53!/52!*(3/805)*(802/805)^52 = .16266... = 16.27% (roughly)
q2: P(X) = 16!/15!*(3/805)*(802/805)^15 = .05637... = 5.64% (roughly)
q3: P(X) = 5!/4!*(1/156)*(155/156)^4 = .03123... = 3.12% (roughly)
q4: P(X) = 2!/1!*(1/48)*(47/48)^1 = .04079... = 4.08% (roughly)
q5: P(X) = 1/16 = .0625... = 6.25%
NOTES: It is important to note that you can't simply add these percentages (except for q5 with 1 trial) for the number of probability because they don't are not linear probabilities when you have more than 1 attempt for involved reasons . The results for 1 BH don't tell the full story as it is much easier to get multiple times in the q3 without trying for q4 or plus. This applies only to the aforementioned battle because number of rounds, hits, and slots filed will differ but the odds might remain more or less accurate for different battles.
5- Conclusion
I am actually surprised the odds were so high in my favor which just pisses me off more! As we can see the best odds are unsurprisingly to get q1 equipement because of the volume given out but I learnt that it is effective to either go for BH medals, top 3, or a high volume of top 10s. In reality, maximizing top 10s is the best way to get equipement because it is relatively easy to repeat but the top 3 and top 1 also have good odds.
Please note I have never taken a stat course and undoubtedly made mistake(s)-- feel free to comment my mistakes and I will try to correct it as well as any other abnormalities you've noticed in equipement distribution probability.
Until next time,
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