Is the expected waiting time for a block always 10 min?

[DeathAndTaxes]

While the expected time between blocks is 10 minutes the time until the next block is unknown.  The expected waiting time for a block in isolation is not very useful because it doesn't tell you the whole story instead it is more useful to look at the distribution of possible values and use probability to provide a level of confidence of how long the next block will take.   Often the question that is behind the question asked is "What is the probability that my txn will be confirmed in the next x minutes?"

Consider these two questions:
1) Right now what is the probability that the next block will be found in 10 minutes or less?
2) What is the length of time such that there is a 50% probability that the next block will be found in that amount of time or less?

Some people incorrectly think the answers are 50% and 10 minutes respectively but that isn't correct.  While the expected block time has a mean of 10 minutes the distribution is not linear.  We can use a CDF (cumulative distribution function) to graph the probability of the next block being found in the next x minutes.

CDF (Bitcoin Blocks = mean of 10 minutes)
50%    6.9 minutes   
63%   10.0 minutes
80%   16.9 minutes
90%   22.8 minutes
99%   42.0 minutes

So the answer to the two questions above are 63% and 6.9 minutes respectively.  Some rounded easy to remember numbers as a rule of thumb 50% chance of of txn being confirmed in <7 minutes, 75% chance of being confirmed in in <15 minutes, 90% in less than a half an hour, and >99% in an hour.   As others have pointed out that is always time from 'now'.  If I submit a txn there is a 50% chance the next block will be found in less than seven minutes but if two hours pass the probability from that point is still 50% chance in the next seven minutes.

All this assumes hashrate is constant and actual difficulty matches the expected difficulty for the hashrate.

[odolvlobo]

For every block that takes an hour, there should be one that takes a few seconds.

True, but...
If you "suffer through" an unusually long streak (high wait time for blocks, dice rolls under 7, etc), then at some point you have to get fast blocks or high dice rolls in order for the stats to normalize. Is it accurate to say the dice have long-term memory?  

The average time between blocks has been about 10 minutes. There is no mechanism to guarantee that the future average time between blocks will be 10 minutes. However unlikely it might be, the average time between the 2016 blocks in a difficulty period could be 1 hour or 30 seconds. If the first 1008 blocks averaged 20 minutes each, there is nothing that will make the last 1008 blocks average 0 minutes in order to maintain the expected 10 minute average. Statistics describing the past can be extrapolated in order to predict the future, but they can't affect the future.

FYI This is related to the fallacy of the concept of "reversion to the mean".

[goose20]

If you "suffer through" an unusually long streak (high wait time for blocks, dice rolls under 7, etc), then at some point you have to get fast blocks or high dice rolls in order for the stats to normalize. Is it accurate to say the dice have long-term memory?  

Not according to the laws of math (i.e. a random number generator could produce a number < X for years without being flawed).

Again the "difficulty adjustment" is used to "try and fix" this sort of problem (in terms of adjusting based upon the history).

For every block that takes an hour, there should be one that takes a few seconds.

True, but...
If you "suffer through" an unusually long streak (high wait time for blocks, dice rolls under 7, etc), then at some point you have to get fast blocks or high dice rolls in order for the stats to normalize. Is it accurate to say the dice have long-term memory?  

The average time between blocks has been about 10 minutes. There is no mechanism to guarantee that the future average time between blocks will be 10 minutes. However unlikely it might be, the average time between the 2016 blocks in a difficulty period could be 1 hour or 30 seconds. If the first 1008 blocks averaged 20 minutes each, there is nothing that will make the last 1008 blocks average 0 minutes in order to maintain the expected 10 minute average. Statistics describing the past can be extrapolated in order to predict the future, but they can't affect the future.

FYI This is related to the fallacy of the concept of "reversion to the mean".

This is interesting. I have done some reading up on 'reversion to the mean' as I've  never heard of the term.

To me, my thinking was along the lines to Bit_Happy. That is, if there is an average x, and for years the number produced is < x,  then there needs to be a period where for years the number produced has to be > x, in order for that average to stand.

Am i understanding the fallacy of the reversion to the mean correctly - the above is not necessary the case?

[CIYAM]

To make it really easy to understand just think of tossing a coin.

If I tossed the coin and it came up "heads" 10 times in a row you might think that there could be something wrong with the coin (i.e. perhaps it has been weighted on one side) and if you didn't think that it was a weighted coin then maybe you now think it is best to "bet it will land on tails".

But if in the next 90 times it landed on "tails" now what are you going to think (presumably you would have changed your bet to "heads" for the same reason as above and now lost a lot) ?

And if in the next 1000 times it landed on "heads" now what are you going to think (presumably you would have decided that it is now "always most likely to land on heads" or perhaps you would have already gone broke) ?

And if in the next 10M times it was basically 50% heads and tails then those "runs" that seemed to mean something actually mean nothing at all (it's just the way that things can happen).

Gamblers fail to grasp this fundamental property of randomness and always tend to think that one event somehow is more likely to have a specific outcome because of previous events.

A fair coin *always* has a 50/50 chance of heads or tails no matter how many times it came up heads or tails in a row before.

It is actually part of our genetic make up to not think rationally about randomness as randomness is generally not very useful for our survival (we usually do better if we "guess right" based upon patterns we follow).

[odolvlobo]

This is interesting. I have done some reading up on 'reversion to the mean' as I've  never heard of the term.

To me, my thinking was along the lines to Bit_Happy. That is, if there is an average x, and for years the number produced is < x,  then there needs to be a period where for years the number produced has to be > x, in order for that average to stand.

Am i understanding the fallacy of the reversion to the mean correctly - the above is not necessary the case?

Your statement is correct, but if the results have been below average, then the average must change, and not the mechanism behind the average.

Here's an example of the "reversion to the mean" fallacy in action:

You flip a coin 100 times, and you get 40 heads and 60 tails. You expected the number heads to be 50%, but you got 40%. Next, you are going to flip the coin another 100 times.

You expect 50 heads for the next 100, so you must expect 90 heads for the total 200, that is 45% of 200 flips, not 50%. If you expect 400 of another 800 flips to be heads, then you expect the total number of heads to be 490, and the average for the 1000 flips to be 49%, not 50%. Eventually with enough flips, the average will approach the expected 50%, but not because something "reverted".

Now, let's assume that you expect the total number of heads after 200 flips to "revert to the mean" of 50%. That means that you expect 60 heads out of the next 100 flips, and not 50. In order for that to happen, then something about coin flipping must change. Otherwise, you must expect 50 heads and not 60.

[goose20]

This is interesting. I have done some reading up on 'reversion to the mean' as I've  never heard of the term.

To me, my thinking was along the lines to Bit_Happy. That is, if there is an average x, and for years the number produced is < x,  then there needs to be a period where for years the number produced has to be > x, in order for that average to stand.

Am i understanding the fallacy of the reversion to the mean correctly - the above is not necessary the case?

Your statement is correct, but if the results have been below average, then the average must change, and not the mechanism behind the average.

Here's an example of the "reversion to the mean" fallacy in action:

You flip a coin 100 times, and you get 40 heads and 60 tails. You expected the number heads to be 50%, but you got 40%. Next, you are going to flip the coin another 100 times.

You expect 50 heads for the next 100, so you must expect 90 heads for the total 200, that is 45% of 200 flips, not 50%. If you expect 400 of another 800 flips to be heads, then you expect the total number of heads to be 490, and the average for the 1000 flips to be 49%, not 50%. Eventually with enough flips, the average will approach the expected 50%, but not because something "reverted".

Now, let's assume that you expect the total number of heads after 200 flips to "revert to the mean" of 50%. That means that you expect 60 heads out of the next 100 flips, and not 50. In order for that to happen, then something about coin flipping must change. Otherwise, you must expect 50 heads and not 60.

Damn you've explained that well. Thanks mate.

[picolo]

If there was a big electrical problem in the US or China, the total hashrate of the network would go down and the time between blocks would likely go up until the next difficulty change.