Block 916467 lasted 51 minutes. I calculate the probability of that.

It turns out that block 916467 lasted significantly more than the expected 10 minutes with 51 minutes total.

I wanted to calculate the probability of that, so here I am sharing it with you.

Bitcoin block mining follows an exponential law with parameter T = 10 minutes. Therefore, the probability that a block lasts longer than rT minutes is

https://preview.redd.it/e7ys0s5jiirf1.png?702&format=png&auto=webp&s=3eb6495bb3aa6fea182b6f1eb5fa04230d819df0

where e = 2.71828.

For 51 minutes, r = 5.1 and so the probability that a block lasts at least 51 minutes is 0.61%. The probability is small but not tiny.

By comparison, the probability that a block lasts longer than two hours (r = 12) is tiny at 0.0006% only.

Interesting fact : even though the average is 10 minutes, the probability that a block lasts more than 10 minutes is not 50%. Using the above formula, it is e^-1 = 36.8% only. And so the probability that a block lasts less than 10 minutes is 63.2%.

You can also use the above formula to calculate the probability for a block to be very short. For example, what is the probability that a block lasts less than one minute ? In that case, r = 0.1, and with the above formula we can calculate that the probability that a block lasts more than 1 minute is 90.48%. And so the probability that a block lasts less than 1 minute is 9.52%.

Similarly, a block has a probability 18.1% of lasting less than 2 minutes. Overall, blocks are much more likely to be much less than 10 minutes rather than much longer than 10 minutes.

I hope you liked the maths.

Before you ask, no, this post was not AI generated. Good luck making chatGPT do decent maths ...

Edit : Soon after this post was posted, block 916472 lasted 73 minutes. Now you know what was the probability of that...