Explaining Bitcoin Mining as a Poisson Distribution

If you ever watched a block chain explorer, such as mempool.space, and waited for the next block to be mined, you would’ve realized that it varies in the time it takes to mine each bitcoin block. However, if you waited long enough and observed many blocks you would realize that the average time for each block to be mined is around 10 minutes.

If we assume that the hash rate is constant the whole time, a new block will be mined once every 10 minutes on average.

The times at which a new bitcoin block is mined could be modeled by a poisson distribution. An important characteristic of a poisson distribution is being memoryless; meaning that whether an event has just occurred or that an event hasn’t occurred in a long time will give us no clue about the likelihood that another event will occur soon.

Because of that, we can always expect the next bitcoin block to be mined in 10 minutes at any given moment. Therefore, if you have already been waiting for 4 minutes now, we still expect the next block to be mined in the next 10 minutes, not 6 minutes, as if you haven’t been waiting at all. This memoryless characteristic also works backwards in time just as well as forwards in time. Therefore it would work the same way I just explained — if you choose a random time, the preceding block will have been mined on average 10 minutes earlier.

Poisson Math

Let’s take a bitcoin miner, Alice, performing a fraction, 0 < p ≤ 1, of the total network hash rate. If we know that the average amount of time it takes for a bitcoin block to be mined is N minutes, then we know that Alice will take an average t = N/p minutes to mine a block.

Now let Tₙ represent a random variable of the time between each of the blocks mined by Alice (waiting time). Due to the pseudo-random properties of the hash function, mining is a Markov process, being memoryless. Which means that one event in the process does not effect any other event and are both statistically identical.