The BIP39 wordlist, comprising 2048 predetermined words, serves the purpose of generating mnemonic or seed phrases for cryptocurrency wallets. Understanding the probabilities involved in this scenario is vital for evaluating the security and uniqueness of these seed phrases. Let’s delve into the probabilities linked to matching varying numbers of words in the BIP39 wordlist, while also discussing the underlying assumptions and factors that can impact the actual probabilities.

1. Matching one word:

Let's consider the probability of two individuals selecting the same word from the BIP39 wordlist. Since there are a total of 2048 words on the list, the probability of two people choosing the same word is given by the fraction 1/2048.

2. Matching two words:

In this scenario, we need to determine the probability of two people randomly selecting the same two words from the BIP39 wordlist. The likelihood of selecting the first word is 1/2048, and the probability of selecting the second word is also 1/2048. To find the overall probability, we multiply these individual probabilities together: (1/2048) * (1/2048) = 1/4,194,304.

3. Matching three words:

Similar to the previous scenario, the probability of two individuals randomly selecting the same three words is calculated by multiplying the individual probabilities of selecting each word: (1/2048) * (1/2048) * (1/2048) = 1/8,589,934,592.

4. Matching all twelve words:

The most significant scenario involves determining the probability of two people randomly selecting the exact same 12-word seed phrase from the BIP39 wordlist. Considering that each word has a probability of 1/2048, we can raise this probability to the power of 12 to account for all twelve words: (1/2048)^12 = 1/2,176,782,336,000,000.

Calculating the probabilities associated with matching 12-word seed phrases from the BIP39 wordlist provides valuable insights into the uniqueness and security of these phrases. The calculations demonstrate that as the number of matching words increases, the probability decreases exponentially. However, it is crucial to acknowledge that these calculations are based on the assumption of truly random and independent word selections. In practice, there might be non-randomness or biases in the word selection process, which can impact the actual probabilities.