extremely long blocks - what are the chances?

[organofcorti]

I think the three and four block sequence probabilities are negative binomial, and if so the median values will be ~ 25 minutes for 3 blocks and ~ 34 minutes for four blocks solved, which also look about right.

If we treat time as continuous it will be the continuous version of negative binomial, which is the Erlang distribution.

I'm fairly certain we should treat time as continuous

With that correction the median time for three blocks to be solved is 26.7 minutes and for four blocks to be solved 36.7 minutes.

[1714101900]

1714101900

[John Tobey]

The probability of a block being solved after 10 minutes: 50%, or 1/2

I might be doing something wrong, but it seems my data shows that 65% of block have been solved in less than 10 minutes. The difference seems a bit too large to be explained by our "faster mining", no?

No, the probability after 10 minutes at constant hashrate would approach, IIRC, 1 - exp(-1), not 50%.  The average time would be 10 minutes, but not the median as 50% would imply.  The long tail at the high end pulls the average above the median.

1-exp(-1) is c. 63%, a little less than your number, as one would expect given the history of increasing network speed.

[molecular]

The probability of a block being solved after 10 minutes: 50%, or 1/2

I might be doing something wrong, but it seems my data shows that 65% of block have been solved in less than 10 minutes. The difference seems a bit too large to be explained by our "faster mining", no?

No, the probability after 10 minutes at constant hashrate would approach, IIRC, 1 - exp(-1), not 50%.  The average time would be 10 minutes, but not the median as 50% would imply.  The long tail at the high end pulls the average above the median.

1-exp(-1) is c. 63%, a little less than your number, as one would expect given the history of increasing network speed.

Yep. Isn't it nice when empirical data matches up with the mathematical model so nicely? It might not be a big deal for most, but it never fails to make me feel deeply content.

Thanks for chipping in John and thanks again for bitcoin-abe.

[deepceleron]

....
The average time would be 10 minutes, but not the median as 50% would imply.  The long tail at the high end pulls the average above the median.

1-exp(-1) is c. 63%, a little less than your number, as one would expect given the history of increasing network speed.

These two statements appear to be in dissonance, but are both true:

• The average time to solve a block is 10 minutes
• Half the blocks will be solved in under 7 minutes

Obviously easy to confuse the two when calculating things.

So the chance a block won't be found after 10, 20 minutes, etc. (and the 1 in x chance)

Code:

>>> for x in range(10,131,10) :
	print x, '%2.5f%%' % (100*math.exp(-x/10)), '%1.2f' % (math.exp(x/10))

10 36.78794% 2.72 (average block length)
20 13.53353% 7.39
30 4.97871% 20.09
40 1.83156% 54.60
50 0.67379% 148.41
60 0.24788% 403.43
70 0.09119% 1096.63
80 0.03355% 2980.96
90 0.01234% 8103.08
100 0.00454% 22026.47
110 0.00167% 59874.14
120 0.00061% 162754.79
130 0.00023% 442413.39

And after how many minutes will half of blocks, 25% of the blocks, etc. not be solved:

Code:

>>> for x in [ 10*math.log(2), 10*math.log(4), 10*math.log(8), 10*math.log(16), 10*math.log(32), 10*math.log(64)]:
	print x, '%2.5f%%' % (100*math.exp(-x/10)), '%1.2f' % (math.exp(x/10))

   
6.9314718056 50.00000% 2.00 (median block length)
13.8629436112 25.00000% 4.00
20.7944154168 12.50000% 8.00
27.7258872224 6.25000% 16.00
34.657359028 3.12500% 32.00
41.5888308336 1.56250% 64.00


And every day we can reasonably expect a block longer than 49 minutes, every week one longer than 69 minutes, every month 83 minutes:

Code:

>>> for x in [ 10*math.log(144), 10*math.log(144*7), 10*math.log(144*365.25/12)]:
	print x, '%2.5f%%' % (100*math.exp(-x/10)), '%1.2f' % (math.exp(x/10))

   
49.6981329958 0.69444% 144.00
69.1572344863 0.09921% 1008.00
83.8548870042 0.02282% 4383.00

I'm curious to see how an integrated histogram of actual block data matches up with this.

Of more significance, if the hash rate temporarily halves (miners quit, pool DDOS, etc) all times double.

[deepceleron]

This seems like the best place to make this available, a CSV dump of block times and durations.
Blocks 0 - 216538, in this format:

Code:

blocknum,epochtime,blocksec,datetime
0,1231006505,0,2009-01-03T18:15:05Z
1,1231469665,0,2009-01-09T02:54:25Z
2,1231469744,79,2009-01-09T02:55:44Z
3,1231470173,429,2009-01-09T03:02:53Z
4,1231470988,815,2009-01-09T03:16:28Z
5,1231471428,440,2009-01-09T03:23:48Z
6,1231471789,361,2009-01-09T03:29:49Z
7,1231472369,580,2009-01-09T03:39:29Z
8,1231472743,374,2009-01-09T03:45:43Z
9,1231473279,536,2009-01-09T03:54:39Z
10,1231473952,673,2009-01-09T04:05:52Z
11,1231474360,408,2009-01-09T04:12:40Z
12,1231474888,528,2009-01-09T04:21:28Z
13,1231475020,132,2009-01-09T04:23:40Z
14,1231475589,569,2009-01-09T04:33:09Z

http://we.lovebitco.in/blocktimes.zip

(space reserved if I make some histograms or conclusions with data...)