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If network would include 2 mining pools (pool_1 and pool_2) with the hashpower of 49% and 51%,respectively, then wo want to calculate the probability that in 10 minutes, exactly 2 blocks would be generated such that two blocks are not generated at the same time and the first block is generated by pool_1 with hashpower of 49% and second block is generated by pool_2 with hashpower of 51% ?

(1) It is important for us to know which pool generates the first block and which one generates the second block, so, we want to know the probability that the first block is generated by which pool (regarding to their hashpower)

(2) we assume that pool_2 is honest and pool_1 is selfish and

(3) We know that Bitcoin network on average generates one block per 10 minutes.)

(According to solution that is proposed by Murch here : How can we be sure that a new block will be found?)

Thanks

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    I understand the scenario you're suggesting, but I don't understand what your question is. – Nick ODell Sep 19 '16 at 20:19
  • @ Nick ODell, I modified question. – Questioner Sep 19 '16 at 21:36
  • I think this may be an XY-Problem. Perhaps it would be helpful to explain why you want to find out about this case? E.g. what's the underlying question about selfish mining that you're interested in? – Murch Sep 19 '16 at 22:16
  • @Murch, In fact, I determined which on is honest and which miner is selfish, since you said "the network is fundamentally broken then: pool_2 can just ignore all blocks by pool_1 and still produce the longest chain by itself" but in general the question is that the probability that which one can generate the first block is depended on their hashpower? According to Nate Eldredge answer it is not denpended, but I think it is more logical that the miner with more hashpower has more chance to generate the first block, I'm wrong? – Questioner Sep 19 '16 at 23:01
  • That's not what Nate said. Of course the chance is bigger that the first block is produced by the pool with the bigger hashrate. And I don't think you understood what I was suggesting. – Murch Sep 19 '16 at 23:42
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In your example the two pools have very similar hash power, so for simplicity let's assume they are both exactly 50%. The difference between 49%, 50%, and 51% will not significantly affect the answers.

The probability that the first pool finds at least one block in 10 minutes is roughly 1 - exp(-1/2), about 39%, and the same for the second pool. The two pools operate independently, so the probability that both events happen is simply the product of their probabilities, i.e. (1 - exp(-1/2))^2, which is about 15.5%. By symmetry, it's about equally likely for either pool to find the first block. So the probability that this happens and moreover Pool 1 finds the first block, is half of this: 7.75%.

  • Thank you for the response. So, do you mean the prob. that the first block is generated by pool_1 is equal to prob. that the first block is generated by pool_2 ? This is not related to their hash power ? consider for eample 99 % hashpower for pool_1 and 1% hashpower for pool_2. – Questioner Sep 19 '16 at 21:25
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    @sas: They're not exactly equal, but I rounded them both to 50% hashpower to make the computation simpler. You may compute the exact number as an exercise; it should not be substantially different. Obviously if it were 99% and 1% this would not be a good approximation. – Nate Eldredge Sep 19 '16 at 21:46
  • @ Nate Eldredge, Thank you. In fact, what I'm looking for is prob. (pool1[selfish] , pool2[honest]) and my question is that if hashpower does not affect probability that which one can generate the first block, thus if for example we change their hashpower such that pool1 : 10% and pool2 : 90%, in this case the probability that the first block is generated by pool1 is equal to probability that the first block is generated by pool2 ? – Questioner Sep 19 '16 at 22:26
  • No, of course they are not equal in that case. You'll have to redo the computation with those numbers. I haven't the time to do it myself right now. – Nate Eldredge Sep 19 '16 at 23:52
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    @sas: Here I am allowing for the possibility that additional blocks are generated during the same 10 minutes. In any case you would not multiply by 18%; this does not make sense as these are not independent events. I am afraid this comment thread is not the place for a course of probability theory. – Nate Eldredge Sep 20 '16 at 13:06

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