# Mining Probability

Assuming two miners have a (1/x) chance of finding a block over some time period, is the probability of both miners finding a block within that period simply (1/x^2)?

More generally, would n miners with equal hash rates have a (1/x^n) chance of all finding blocks within some time period of each other?

This is for a research project where mining probabilities are being factored into a simulation, and I'm not sure if certain configurations are eliminated when the first miner finds a block (ie they would be dependent events)

In addition to what @JBaczuk wrote, you may be interested to learn more about orphan blocks, which are created when two miners find a valid block within the latency period of block transmission across the network.

In practice, the amount of time passing between the finding of a valid block, and that block propagating across the network is very, very small, as much work has been done to optimize the network for fast block transmission (see: the FIBRE network). But within that small window of time, there is a small chance that two miners would find competing blocks.

Keep in mind: if a miner has ~10% of the hashrate, then they have a ~10% chance of finding a new block within the next 10 minutes. The chance of them finding a block within a time period of a couple seconds would be much smaller. The chance of two miners finding competing blocks within those couple of seconds would become even much smaller still!

• Would it make sense to use a Poisson distribution here? The idea is that you could assume a block is found every 10 minutes on average, so the chances of finding two blocks within 60 seconds is something like 0.45%. Of course this seems like a pretty generous upper bound assuming a network without any adversarial nodes purposely not propagating solutions. And I guess there is also the external consideration that this refers to two blocks in one minute, and not necessarily two blocks that have the same parent (ie a fork) – user97236 Jul 25 '19 at 23:55
• @user97236 yes, mining can be modelled as a Poisson process. I'd agree about the external consideration of whether two blocks being found have the same parent or not, but this externality will not affect the probabilities of finding some block, just the higher-level condition of block competition/validity. – chytrik Jul 26 '19 at 2:33

Assuming two miners have a (1/x) chance of finding a block over some time period, is the probability of both miners finding a block within that period simply (1/x^2)?

More generally, would n miners with equal hash rates have a (1/x^n) chance of all finding blocks within some time period of each other?

I believe these are both correct assuming the miners are working independently of each other.

In practice, however, as soon as a miner finds a block, it is in their best interest to publish that to the network as soon as possible to prevent another from publishing a block before them. So, there is a certain amount of time where the new block will propagate across the network after which a miner will not attempt to mine that old block anymore (usually seconds), so network latency has an effect here.