Calculating Champ Probabilities

MadScientist

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 5-star 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/1Q4qzZHC6r0bEr95IptXr47-RLDCP-VChiqEkoGNlORQ/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 5-star 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.

Yoik

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 pull