# The probability of block generation by two miners in a special case?

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

• 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

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%.