Calculating Champ Probabilities
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MadScientist
Posts: 317 Mover and Shaker
I wanted to know the probability to be able to champ a character after getting a certain number of random covers so I did some calculations. The results were quite interesting so I thought I share them with you.
To be able to answer my question I calculated the probability for each possible cover distribution (from 000 up to 553) when staring from a given distribution and adding various numbers of random covers. For example, if you start with zero covers, the distribution after adding 13 random covers is
Distribution  Probability + 500  0.00000188 (= 3 * (1/3)^13) 510  0.00004892 (= 6 * 13 * (1/3)^13) 511  0.00029354 (= more complicated ) 520  0.00029354 521  0.00322896 530  0.00107632 522  0.00807239 531  0.01076319 540  0.00269080 532  0.04843435 541  0.02421718 550  0.01130135 533  0.06457913 542  0.09686870 551  0.06135018 543  0.22602697 552  0.13561618 544  0.16952023 553  0.13561618
So the probability to be able to champ a character after getting exactly 13 covers is the probability to have a 5/5/3 or a 5/4/4 cover distribution which is 0.16952023 + 0.13561618 ~ 30.51%. If we continue this exercise, we can calculate the expected number of random covers required to champ a character, which is 15.36. If these random covers are 5star covers from latest legends tokens, which give a cover for specific character with a probability of 1/20, you will need 15.36 * 20 = 307.2 tokens on average (this number is remarkably close to the 300 required tokens widley circulated).
If you already have some covers and/or enough shards to buy one or more covers, the problem is a little bit different. For example, if you have no covers but enough shards to buy 2 covers, you only need 11 covers to be able to champ the character (11.80 covers on average)
All results can be found in this table: https://docs.google.com/spreadsheets/d/1Q4qzZHC6r0bEr95IptXr47RLDCPVChiqEkoGNlORQ/edit?usp=sharing
How to read this?
 The first column "Initial cover distribution" is the covers you start from. It is always in descending order since there is no difference between a 1/0/0, 0/1/0, or 0/0/1 distribution. No covers is denoted by the value "0". All possible covers distributions were calculated.
 The second column "Number of initial covers" is sum of covers from the first column.
 The third column "Initial shard covers" is the number of covers you can buy via shards. This reduces the number of covers needed to champ the character.
 The fourth column "Expected random covers" is the average number of random covers required to champ the character. You will find the value 15.36 I mentioned above in the first row, for example.
 The fifth column "Stddev random covers" is the standard deviation of the average. If you don't know what a standard deviation is, just ignore it. If yes, you know what this means
 The following columns "p(<n>)" are the probability to be able to champ the character after getting *exactly* n covers. You will find the number 0.3051 I mentioned above in column "p(13)" in the first row, for example. The number n goes up to 19.
Let's look at some numbers:
 In the days before shards, you had to champ a new 5star character starting from zero covers using only random covers gained from legendary tokens. On average, you need 15.36 random covers (~307 tokens).
 After shards and CL10 were introduced, you could gain 1, 2, or 3 covers from shards. Then you need 13.40, 11.80, or 10.45 covers on average (~268, 236, or 209 tokens).
 Starting with Gargantos, it looks like you can start with a 1/1/0 character and will gain addional 1, 2, or 3 covers from shards. In this case you need 11.26, 9.69, or 8.36 covers on average (~225, 194 or 167 tokens)
 My Moon Knight has currently 3/2/1 covers and 1375 shards. I will reach 3 shard covers after his next featured event. I will only need 4.20 covers on average to be able to champ him.
2
Comments

This is very interesting to me thank you. I have approx 270 tokens ready for new thor to go into the packs. Ive got 2 covers already thanks to the recent team boss battle. Maybe my 270 tokens will be enough to cover her if i explicitly make her my unique champ/shard toon for that pull0
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