One more interesting observation: as already mentioned, the probability of the next block not being found within 30 minutes, is e-30/10 ≈ 4.98%
However, when this occurs, e.g. no block as been found for 30, or 40, or 60 minutes, then at that point the next block is still expected to be found 10 minutes after that.
In other words, at any given moment (no matter how short or how long ago the last block was found!) you can expect the next block to still take 10 more minutes.
I think that because finding the winning hash of a block is a Poisson random process, no matter how long it took to find a block, the expected value of time to find the next block is 10 minutes,
but as time goes by since the last block was found, it is more and more likely that a block will be found. The probability that a block will be found in less than time "t" is given by 1 - e
-Lambda*t,
where Lambda = 1 / Expected Block Interval Time; (for Bitcoin, Lambda = 1/10 minutes or 1 / 600 seconds)
So for example, if you want to be 99.9% sure when the next block will be found, you can set 0.999 = 1 - e
-(1/10)*t and solve for t.
0.999 - 1 = - e
-(1/10)*t // subtract 1 from both sides
1 - 0.999 = e
-(1/10)*t // multiply both sides by -1
ln( 1 - 0.999) = -(1/10)*t // take natural logarithm of both sided
ln( 1 - 0.999) * 10 = -t // multiply both sides by 10
t = 69.08 minutes, = about 1 hour 9 minutes 5 seconds
So, you can say, that after a block has been found, it is 99.9% certain that the next block will have been found within the next hour nine minutes and 5 seconds.
If you want to be 99.999% sure, the same calculation yields 115.13 minutes or 1 hour 55 minutes and 8 seconds
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