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What is the probability distribution of solving a block, given the same difficulty.

So if I try to mine multiple times using the same difficulty, is it normal distribution with mean of 10 minutes? What is the variance? Or is it some other distribution? Or maybe even deterministic chaos?

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  • It is a poisson distribution. Every attempt to form a block has the same probability of success and the number of attempts per second is roughly constant.
    – David Schwartz
    May 12, 2014 at 23:47
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    @DavidSchwartz: No, it's a Poisson process. Which means that the number of blocks in an interval follows the Poisson distribution - but the time between blocks, which the OP asked about (I think), follows the exponential distribution.
    – Meni Rosenfeld
    May 13, 2014 at 9:14
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    @MeniRosenfeld The question is sufficiently vague that it could be asking either about the expected number of blocks found or about the expected time between blocks found. They follow two different distributions.
    – David Schwartz
    May 13, 2014 at 17:34

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The time between consecutive blocks follows the exponential distribution, with mean (roughly) 10 minutes. This means that the variance is 100 minutes^2.

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  • so standard deviation is 10? So its just as likely to take 1 minute vs 19 minutes?
    – jaybny
    May 12, 2014 at 18:43
  • bitcoin.stackexchange.com/questions/4690/…
    – jaybny
    May 12, 2014 at 18:58
  • @jaybny: Yes, the standard deviation is 10 - however, the distribution is not symmetric, so 1 minute and 19 minutes have different probability densities.
    – Meni Rosenfeld
    May 13, 2014 at 9:12
  • im now asking if the probability that it takes exactly 1 minute or exactly 19 minutes the same? since they are both the same STD from mean?
    – jaybny
    May 13, 2014 at 13:39
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    (...continuing from previous comment) some probability. These trials are all independent. So the time to find a block can be arbitrarily long (because I can just be unlucky and fail again and again and again). But the probability decreases exponentially: The probability to find a block in the 4th minute is the probability to fail in the 1st and fail in the 2nd and fail in the 3rd and succeed in the 4th. Because of the extra conditions, this has less probability than just succeeding in the 1st minute. This results in the en.wikipedia.org/wiki/Exponential_distribution.
    – Meni Rosenfeld
    Mar 15, 2016 at 14:21
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The expected time (mean) for a new block is of course 10 minutes, assuming constant hashrate, and no block propagation time.

The tricky part is that there is no such thing as a point in time. You can only ask only for an interval.

Let's illustrate this. First it is important to not fall for the Gambler's fallacy. Luck has no "memory". Thus if no block has been found (or if a block has just been found), what is the chance of a block being found in the next minute? The easy answer would be 1/10 = 10%. Or in the next second? 1/600 = 0.16667%. But this is not quite true.

If you ask how often a block is found before or after 10 minutes, you are asking for the cumulative distribution function (CDF) of the exponential distribution.

We can use Wolfram Alpha to plot this:

cdf exponential distribution λ=1/600

How many blocks are found after the mean (600 seconds)?

exp(-600/600) = exp(-1) ~= 36.788%

How many blocks are found before the mean?

1-exp(-1) ~= 63.212%

How many blocks are separated by more than 1, 2, 5, 10, 20, 30, 60 minutes?

exp( -1/10) ~= 90.484%
exp( -2/10) ~= 81.873%
exp( -5/10) ~= 60.653%
exp(-10/10) ~= 36.788%
exp(-20/10) ~= 13.534%
exp(-30/10) ~=  4.979%
exp(-60/10) ~=  0.248%

So what is the chance of finding a block in the next second/minute?

1-exp(-1/600)  ~= 0.16653%
1−exp(−60/600) ~= 9.516%

Bonus: How many blocks are found between 5 minutes an 20 minutes?

exp(−5/10)−exp(−20/10) ~= 47.120%
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