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 Author Topic: Probability of Next Block Within 10 Minutes  (Read 2895 times)
AnonBitCoiner
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 January 02, 2015, 02:54:09 AM

It's a Poisson process, with a mean of 10 minutes. So...

What is the probability that the next block will be found within 10 minutes from now?
63.212%

Or within 1 minute?
9.516%

Or 10 seconds?
1.653%

Or 1 second?
0.167%

Holy guacamole. How to tell a BitCoiner with statistics background versus a newblet.

Take a bow, sir.

How much information is typically in a block? I'm still a novice

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dbkeys
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 January 30, 2015, 11:02:54 PM

It's a Poisson process, with a mean of 10 minutes. So...

What is the probability that the next block will be found within 10 minutes from now?
63.212%

Or within 1 minute?
9.516%

Or 10 seconds?
1.653%

Or 1 second?
0.167%

I'll join in the applause ... and then ask, could you please show how you arrived at these answers ?  I would like to know how to apply this to other crytpos, which aim for different block interval times.

OROBTC
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 January 31, 2015, 05:20:16 AM

...

Poisson distribution, but approximates its continuous cousin the exponential distribution.  For me it is easier to think in terms of the exponential.

When I am waiting on a confirmation for a transaction just done, the rule of thumb I use is (very similar math to Foxpup's) is that a confirmation (let's just use a simple rule of a solved block = a confirmation, not always true) will come in a median of the 6.32 mins:

10 (minutes) - 10(1/e) = 10 - 10(0.3678) = ~ 6.32 minutes (median) even with the designed 10 minute average.  Where "e" is that pesky number that shows up in the most unusual places (and is approximately 2.71828).

What seems to happen to me a lot is that confirmations (time until next solved block) seem to take 20 - 25 minutes rather often...:

P(> 20 mins) = 1/e^2 = some 13.5%, wow, I'm unlucky.  So don't look for me in Vegas...

OROBTC
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 January 31, 2015, 05:41:18 AM

...

Here's a chart I did for curiosity back on Nov 27 of last year tracking the times between blocks (info from blockchain.info):

Where minutes between blocks are the Y-Axis and the 112 or so consecutive blocks on the X-Axis.

The mean and standard deviation of that data set (both in minutes):

9.92920354
7.700552872

(Sorry I could not figure out how to post the image right into here)

EDIT: My understanding of the exponential distribution (probably the Poisson as well) is that the standard deviation should (approximately in reality) equal the mean, in this data set that is not true...  Please correct me if I am wrong, smile,,,

Jace
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 January 31, 2015, 10:23:07 AM

There's no probability actually. It just averages around 10 minutes per block.
Of course there is.

The block time is actually an exponentially distributed variable, which is a continuous process (rather than a poisson distribution which is discrete, although they're closely related) with mean = 10 minutes.
This implies the probability of the next block being found in t minutes from now, is 1-e-t/10 (with e≈2.71828183 or ex = exp(x)).

The reason the actual mean is probably a bit lower than 10 minutes, is because the total hash rate is almost continuously increasing, whereas the difficulty (which is supposed to keep the mean at 10 minutes) is based on the previous 2016 blocks.

Wouldn't a number closer to 50% make sense for the 10 minute mark?
Of course not. 10 minutes is the mean, not the median. The median (the point where 50% are higher and 50% are lower) is 6 minutes and 56 seconds. It should be expected that the median will be much smaller than the mean because there is a lower bound to the block time (zero) but no upper bound. It is possible to have a block time to be 20 minutes longer than average, but it is obviously not possible for the block time to be 20 minutes shorter than average (because it would be less than zero). Thus there a few very long block times that aren't (and can't possibly be) matched by very short block times, and this skews the average.
Very well explained sir

could you please show how you arrived at these answers ?  I would like to know how to apply this to other crytpos, which aim for different block interval times.
In general, the probability of the next block being found in t amount of time, when the average block time is b, is 1-exp(-t/b) (you can express t and b in minutes, seconds, fortnights, whatever you prefer).

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Jace
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 January 31, 2015, 10:28:58 AM

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.

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Bitcoinexp
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 January 31, 2015, 11:39:37 AM

It's a Poisson process, with a mean of 10 minutes. So...

What is the probability that the next block will be found within 10 minutes from now?
63.212%

Or within 1 minute?
9.516%

Or 10 seconds?
1.653%

Or 1 second?
0.167%

How'd you arrive at those values if you mind me asking? Anyone can explain?

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Billbags
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Brainwashed this way

 January 31, 2015, 06:50:24 PM

Bitcoin's designed to put out a new block every ten minutes. And it's going good. Difficulty controls it and most of the blocks are mined 10 minutes apart.
There are a few exceptions when two blocks are found very fast, in a row, and that's just called luck.

There's no probability actually. It just averages around 10 minutes per block.

This could be an interesting read for you too: https://bitcointalk.org/index.php?topic=135982.0

Great topic....we don't get good topics like this much anymore.

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PaulPierce
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 January 31, 2015, 08:30:34 PM

well..somehow most of my transactions take more than 10 mins for confirmations.. no clue y is that..but it seems like every time..over 10 mins.! very occasionally under 10 mins..!! I put fee as 0.0001..

RawDog
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 January 31, 2015, 09:15:02 PM

It's a Poisson process, with a mean of 10 minutes. So...

What is the probability that the next block will be found within 10 minutes from now?
63.212%
....

Wouldn't a number closer to 50% make sense for the 10 minute mark?
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R2D221
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 February 01, 2015, 01:01:37 AM

well..somehow most of my transactions take more than 10 mins for confirmations.. no clue y is that..but it seems like every time..over 10 mins.! very occasionally under 10 mins..!! I put fee as 0.0001..

I'd say it's just luck.

An economy based on endless growth is unsustainable.
dbkeys
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 February 01, 2015, 02:40:16 AM

It's a Poisson process, with a mean of 10 minutes. So...

What is the probability that the next block will be found within 10 minutes from now?
63.212%

Or within 1 minute?
9.516%

Or 10 seconds?
1.653%

Or 1 second?
0.167%

How'd you arrive at those values if you mind me asking? Anyone can explain?

I asked the same question, and it led me to read about Poisson random processes, and the exponential distribution, which is the related time-interval probability density function. To get the probability that a block will be found within whatever amount of time, call it "t", you integrate the probability density function from "t" to infinity to get the probability that it takes longer than "t" and subtract that from 1 to get the probability that it takes less than "t".
Call the Expected block interval time "Beta" = 10 minutes or 600 seconds. (It does not matter, as long as the same units are used throughout), and call the inverse of the expected block interval time "Lamda".
The probability density function is:   Lambda * e -Lambda * t
The probability that "x" (the time to find a block) will be larger than "t"  =  Integral(from "t" to Infinity) of Lamda * e - Lambda * t dx,  which fortunately, simplifies to e-Lamda*t
The probability that "x" will be less than or equal to "t" is therefore = 1 - e-Lamda*t
This is the equation that gives these values:
P(x <= 10 minutes) = 0.63212 or 63.212 %
P(x <= 1 minute)    = 0.09616 or   9.616 %
P(x <= 10 seconds) = 0.01653 or 1.653 %
P(x <= 1 second)    = 0.001665 or 0.1665%

dbkeys
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 February 01, 2015, 03:07:29 AM

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
-

runpaint
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 February 01, 2015, 03:26:48 AM

Probability of BTC price over \$1000 within 11 months is 2,199,114,855/700,000,000 (which comes out to exactly 3)

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R2D221
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 February 01, 2015, 04:40:56 AM

Probability of BTC price over \$1000 within 11 months is 2,199,114,855/700,000,000 (which comes out to exactly 3)

Probability goes from 0 to 1. What you say makes no sense.

An economy based on endless growth is unsustainable.
runpaint
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 February 01, 2015, 10:27:11 AM

Yeah so if the probability was 0, and now it's 3, then that's 3 times more likely (p = 0 x 3 = 3.141592654)

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shorena
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No I dont escrow anymore.

 February 01, 2015, 10:43:31 AM

Probability of BTC price over \$1000 within 11 months is 2,199,114,855/700,000,000 (which comes out to exactly 3)

Probability goes from 0 to 1. What you say makes no sense.

Just use the ignore button.
runpaint
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 February 01, 2015, 11:00:54 AM

You're right, he must be trolling.  Nobody could have misunderstood my comment that badly, so I shouldn't have responded to him as if he was a legitimate poster.

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R2D221
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 February 01, 2015, 07:18:52 PM

Yeah so if the probability was 0, and now it's 3, then that's 3 times more likely (p = 0 x 3 = 3.141592654)

Now I don't know if you're being sarcastic or a troll or what. 0 x 3 = 0. Simple math.

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runpaint
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 February 01, 2015, 08:56:50 PM

Yeah so if the probability was 0, and now it's 3, then that's 3 times more likely (p = 0 x 3 = 3.141592654)

Now I don't know if you're being sarcastic or a troll or what. 0 x 3 = 0. Simple math.

It's a humour-joke, referencing a previous post in this topic

BARR - Burning Altcoins for Redemption and Reduction - First Coin Using Multi-Proof-of-Burn - http://barr.me - Raising the BARR by Reducing Supply - Absorbing and Destroying Entire Altcoin Networks
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