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We know that Bitcoin network on average generates one block per 10 minutes.
Is there a formula to compute the probability that the entire network generates at least two blocks within 10 minutes by using current hashrate of the network and current difficulty level ?
Thanks
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Since current hashrate and current difficulty level should be matched up usually, we can disregard them. (Although, we could easily accommodate them if they are diverging as well.)
Finding blocks is a Poisson Process.
The chance to find at least two blocks in ten minutes is "all cases" excluding the cases where we find 1 and 0 blocks: P(2+) = 1 - P(1) - P(0)
P(0) ≈ 0.368
P(1) ≈ 0.368
→ P(2+) ≈ 0.264 = 26.4%
I've described the formula here: How can we be sure that a new block will be found?
Thanks for useful information. Thus, it means that the probability that the network generates exactly 2 blocks in 10 minutes is 36% ? (maybe I'm wrong ...) – Questioner – 2016-09-18T19:21:59.870
@sas: No, exactly two blocks in ten minutes would be 18.4%. – Murch – 2016-09-18T19:23:07.403
Thank you very much for your helpful information. Now the question is that if network would include 2 mining pools (pool_1 and pool_2) with the hashpower of 49% and 51%,respectively, then the probability that in 10 minutes, 2 blocks would be generated such that the first block is generated by pool_1 with hashpower of 49% ? Thanks – Questioner – 2016-09-18T20:08:49.023
0.184×0.49? However, the network is fundamentally broken then: pool_2 can just ignore all blocks by pool_1 and still produce the longest chain by itself – Murch – 2016-09-18T20:12:24.247
Yes, but we assume that pool_2 is honest and pool_1 is selfish. So, the probability under this condition is 0.184×0.49? – Questioner – 2016-09-18T20:16:53.527
Well, yes, but the second block would be from either pool. – Murch – 2016-09-18T20:22:48.023
And if we want to compute prob. that the first block belongs to pool_1 (selfish) and second one belongs to pool_2 (honest) ? – Questioner – 2016-09-18T20:30:54.450
I don't know how to calculate that exactly from the top of my head, but I would intuitively assume that it is roughly 0.184×0.49×0.51 ≈ 0.046, because we're looking at the cases where two blocks are found in 10 minutes, but only 49% of the first blocks are from pool_1 and only 51% of the second blocks are from pool_2. This disregards disadvantages from relay and validation, though. Perhaps you should rather ask a new question, because comments are not really for extended discussions. – Murch – 2016-09-19T07:01:46.197
Thank you so much. but I think in this case prob. that the first block is generated by pool_1 is equal to prob. that the first block is generated by pool_2. However, I try to ask another question for this. thanks – Questioner – 2016-09-19T19:56:27.307
here is the new question http://bitcoin.stackexchange.com/questions/48667/the-probability-of-block-generation-by-two-miners-in-a-special-case
– Questioner – 2016-09-19T21:40:28.683Let us continue this discussion in chat.
– Murch – 2016-09-19T22:03:53.967