How likely is it that the 12 words that are produce ;when creating a trust wallet or any other wallets that use BIP39 standard for producing pass phrases for your wallet; Duplicate?
The BIP 39 standard uses a list of 2048 English words. You can see all these words in this link:
https://github.com/bitcoin/bips/blob/master/bip-0039/english.txt .
Retrieval words can be made up of different types, such as 12, 18, or 24. Of course, the final Master Seed? is just a string of numbers, and each retrieval word represents a specific numeric code. In fact, because words are easier to write, maintain, and memorize than numbers, the final key is shown to the user in a few words.
BIP39 Standard for creating 12 words have 5 step:
1)Entropy Generation
Entropy is a 128 bit string of 0 and 1 that generate absolutly random.for example suppose that our Entropy string is:
0001101001111101110.......0101110001
2)Checksum Generation
for creating Checksum in this type(i.e for creating 12 pass phrase ) first 128 divided to 32 the result is 4 . then according to SHA256 algorithm first 4 bit of the result of SHA256 algorithm is Checksum for example Checksum is 0001 that add to the end of Entropy and the length of Entropy change from 128 to 132.in our example string change to :
[ ] 0001101001111101110.......01011100010001
3)Split
after this proccess 132 bit string devided to 12 sub string with length of 11 bit
4)To Decimal
every 12 sub string change to decimal numbers that represent th number of word in the list of 2048 words.
5)Mapping and finding every 12 words from list .
This 5 steps is the Bip39 standard process and the result is 12 words pass phrase.
At first it may seem that we want to choose 12 words among of 2048 words ; So the number of cases in which 12 words can be selected from 2048 words is:
2048?2047?2046?2045?2044?2043?2042?2041?2040?2039?2038?2037 ~ 5.27?10^39
and the answer of our question is 1 divided to this number.
but this is wrong......
Because we do not directly choose these 12 words ourselves , but these 12 words are the output of a process called BIP39.So to calculate this probability we have to calculate the number of all the different states that BIP39 can produce these 12 words.
As I said above, BIP39 initially produces a completely random 128-bit string.So to get the answer, we need the number of states that a 128-bit string can be put together and that is simple:
2?2?2?2?2?2..........128 times .......?2 = 2^128
So the probability of duplication of 12 words is 1 divided to 2^128.
For information, let me remind you that the total number of atoms of all objects on Earth is 2^166.
Therefore, the probability of duplication of 12 word is very small and almost impossible.
