Chance of a word repeating THREE times in a BIP39 Mnemonic passphrase? And what about 2 words repeating 2 times each?

Question

So according to the other thread about BIP39 repeating words, the cahnce of a word repeating TWICE is 11.7%.

Now I was wondering, since my passphrase may or may not contain THREE repetitions :O. And then just to finish the chance calculation wizards off, what is the chance that two words are both repeated twice?

We are talking about 24 word passphrases here.

Thanks for teaching us how to do this, or atleast understanding the math :).

Answer 1

The probability of 11.7% is being calculated using the birthday paradox formula.

"The paradox of birthdays is a mathematical problem put forward by Von Mises, who looks for the value N in the problem: In a group of N people there is 50% chance that at least 2 people in the group share the same birthday (day + month). The answer is N = 23, which is quite counterintuitive, hence the paradox."

So for the calculations: -Repetition (Twice): 2 (number of people sharing the same birthday) -Number of words: 24 (number of people in a group) -Number of all BIP39 words: 2048 (number of days in a year)

The result is 0.119 (11.9%). But for THREE repetitions it's getting really low: 0.00043 (0,043%).

Answering the question about two words both repeated twice, would require reworking the formula. I used an online calculator to provide the values.