Explain xkcd: It's 'cause you're dumb.
The comic shows a seven segment display (aka calculator-style numbers) with a countdown. Black Hat explains that it is a countdown to a supervolcano eruption. However, an unfortunately placed picture blocks view of the full display. Due to the form of a seven-segment display, the first digit could be 0, 6, or 8, and several digits are completely blocked by the picture, which Black Hat lazily refuses to move.
The fully visible part starts at 2409, and based on the pace of the scene, it seems to be in seconds. Thus, it is unclear when the eruption might occur. If the obscured digits are all 0's, it could be as soon as 40 minutes. On the other hand, if the obscured digits are '899999', there's another 2.85 million years to go; if they are '000001', we have a little more than 3 years.
The title text: "For all we know, the odds are in our favor" refers to the naive assumption that since we can't see the digits behind the picture, we can treat them as random. If so, the odds are only 1 in 300000 they are all zeroes.
However, we have information about the other digits of the number, and statistical principles such as Benford's law predict it is improbable for the middle 4 digits to be all zero unless the count really is just 2000 something (i.e. all of the higher digits are zero).
- [Counter: ____00002409]
- Cueball: What is that?
- Black Hat: Countdown.
- [Counter: ____00002400]
- Cueball: To what?
- Black Hat: Supervolcano, I think. I forget which one.
- [Counter: ____00002396]
add a comment!
- [Counter: ____00002382]
- Cueball: Maybe we should move that picture?
- Black Hat: Too hard to reach. It's probably fine.
If you assume (with nothing else known), that large numbers have a probability about reciprocal to themselves to ensure a sum/integral of 1, the digits not being zeroes is extremely unlikely.
Whether black hat guy thinks a supervolcanoe eruption is a favourable event or being spared from one is not made entirely clear. Sebastian --22.214.171.124 08:56, 11 January 2013 (UTC)
- I warmly recommend the article harmonic series (mathematics). ;-) --126.96.36.199 13:30, 11 January 2013 (UTC)
- You are right, the harmonic series is divergent. However, the maximal number of digits - which can be possibly displayed - is finite. Which distribution would you suggest? Sebastian --188.8.131.52 19:35, 11 January 2013 (UTC)
- Sebastian, do you know the specific name of the statistical principle you're invoking? I agree, but St.nerol does not, and he has a quick tendency to remove things. One part of it is that you don't know the magnitude of a number, exponential distribution is a more appropriate model than linear. Another part is about the unlikelihood of the middle digits being zero. - Frankie (talk) 21:37, 11 January 2013 (UTC)
- Benford's law is about the probability of certain first digit(s). Sebastian --184.108.40.206 22:34, 11 January 2013 (UTC)
- Hmm... "Benford's law also concerns the expected distribution for digits beyond the first, which approach a uniform distribution". I missed that the first time I read the article. Okay, that covers the essential parts of the argument. - Frankie (talk) 19:43, 12 January 2013 (UTC)
- Come on now Frankie, I'm doing my best. I was just too quick to think that the claim was just another of these casual confusions about probability that non-math people have from time to time. (You know, I haven't rolled a 6 for some time, so now the chances must be pretty high...) I hadn't heard about this very counter-intuitive Benson-principle before, but found this page helpfylly explanatory.
- So, I trust you on this. What I don't understand is, how do we know that Benfords law can be applied to this particular 14 digit number? The time left to an eruption? Also, how could a calculation of the actual probabiliy of the preciding digits being zero or anything else be made? – St.nerol (talk) 22:52, 12 January 2013 (UTC)
- What is more important for this comic than the Benford's law itself, is its underlying condition that many naturally existing numbers are lognormally distributed. And not uniformally distributed. Under that premise we can try do hypothesize about the odds of leading zeroes. Sebastian --220.127.116.11 00:28, 13 January 2013 (UTC)
- The initial timer is a physical quantity, therefore scale invariant, and created by a lognormal distribution (first random experiment). Now there are two possibilities: -- a) BHG specifically got a 14-digit display for the countdown (with the first digit according to Benford's law of course) and the initial timer 14 digits wide. b) The initial timer value possibly was much smaller and it could have been any number which fit on the display. -- Cueball comes in. The shown timer is uniformally distributed within the range below the initial timer (second random experiment). Because of the visible zeroes a) does not seem to be likely and b) would be true, specifically b) with the hidden digits being zero, as the shown zeroes are very unprobable with all large timer values, and the short timer actually is quite probable (lognormal distribution). Is this a valid way to argue for probabilities? Sebastian --18.104.22.168 00:55, 13 January 2013 (UTC)
- It seems legit, but I can't tell, really. But we have no concrete estimation yet (maybe that's too hard). Do you really think that this phenomenon is so strong so that (from the 1 in 30000) it makes the probability for four zeroes higher than for all the other 29999 possibilities together? –St.nerol (talk) 08:45, 13 January 2013 (UTC)
- Another effect is that if the initial counter was small to begin with, it is quite unprobable (with only one supervolcanoe eruption) that Cueball comes in during the run of the counter. I will try to do a calculation example to compare the possibilities with reasonable assumptions. Sebastian --22.214.171.124 08:52, 13 January 2013 (UTC)
- I restructured the last part somewhat. Hope that I didn't screw anything up, and if so, fix it! And it would be very nice if you could also add some more explanation of the math involved! –St.nerol (talk) 19:36, 13 January 2013 (UTC)
- This is a wholly inappropriate accusation to make here. If you have a problem, please put it through appropriate channels. No editor has a perfect score, we all slip up because we're all human. lcarsos_a (talk) 23:49, 12 January 2013 (UTC)
- Assuming that the middle digits are random, the expected value is 1.53 million years. But: If the display is off-the-shelf, it is probably larger than the largest number actually displayed. Maybe the counter started at 1e8, and the next smaller display had only 8 digits. Maybe we should have a look at the statistical distribution of digits in commercially available LED displays ... 126.96.36.199 08:42, 14 January 2013 (UTC)
- I don't think there are displays with that many digits. You have to buy several one digit (perhaps four digits) displays and multiplex them together. 23:56, 15 February 2014 (UTC)
"I forget which one" may be a reference to the 7 known supervolcanoes, or it might be to a list published by the Guardian in 2005 of the top 10 existential threats to life on Earth, which went briefly viral. It included a supervolcano eruption, as well as viral pandemic, meteorite strike, greenhouse gases, superintelligent robots, nuclear war, cosmic rays, terrorism, black holes, and telomere erosion 
I understand how the hidden numbers could mean that a volcano could either erupt very soon or a very long time. But I don't get why this is a joke. Is there something funnny that I am missing? 188.8.131.52 (talk) (please sign your comments with ~~~~)
- It's a joke, because a supervolcano eruption would have a major impact on the earth, and Black Hat has a timer that will tell him when one will occur, but he is too lazy to see whether it will happen soon. 184.108.40.206 (talk) (please sign your comments with ~~~~)
The title-text may be a reference to the line "May the odds be ever in your favor!" in The Hunger Games. I wonder if this might also be a commentary on the foolishness of assuming that a rare event won't happen anytime soon. gijobarts (talk) 19:54, 12 January 2013 (UTC)
The picture could be somewhat symbolic. It could be a sunset or sunrise, like the would could be about to end or not. 220.127.116.11 06:19, 13 January 2013 (UTC)
Benford's Law has no bearing on what any of the covered digits are except the first, and even then it only weakly applies; it only applies to the FIRST digit of natural numbers, and since we can have leading 0's is really doesn't apply. Furthermore, even if it applied to all the digits, the probability distribution on the covered digits is not affected by the shown digits; that's not how probability works. If I flip a coin 10 times and it's heads all ten times, the probability that the 11th flip is still 50/50. -Mike Powers
- Benford's Law shows that with real-life (physical) numbers you cannot just use a 10% probability for each digit. These numbers are not uniformally, but lognormally distributed. That means, there is a smaller tendency to greater numbers than their possible number space would allow. Benford's Law with its relevancy to the first n digits is not directly applicable here, but its general validity contradicts some of the assumptions normally often made. As you see many zeroes in the middle part, the probability is quite high that also the first digits are zero. Here the length of the number has a normal distribution and a short number is about as probable as a long one. And long ones with zeroes in the middle are seldom so it is probably a short number. This would not be the case, if each digit is randomly selected from 0-9. Then the greater probability of longer numbers would cancel out this effect. Sebastian --18.104.22.168 10:07, 3 February 2013 (UTC)
- Regarding the independence of the digits: That is conditional probability. We have a probability distribution for the complete number. In nature this is a lognormal distribution (with suitable parameters regarding the scale; that is why the intention to buy a display with certain width is important). That means zero digits are quite common, as short numbers have much weight. With just creating the digits independently you do not get a lognormal distribution. With four zeroes shown only 1/10.000 of the longer numbers are possible any longer, making them much rarer. To begin with they would need a probability of at least 10.000 as high to counter this effect, but they do not have it (with a uniformal distribution they would have it). Sebastian --22.214.171.124 10:25, 3 February 2013 (UTC)
- If we have initially the same probability for numbers of digit length 1-14 (about 7%): After looking we (partly) know that digits 1 till 4 are non-zero and digits 5-8 are zero. Then numbers of digit length 1-3 have 0% probability, numbers with digit length 5-8 have 0% probability. Numbers with digit length 9-14 have a probability of 0.01% each and numbers with length 4 have a probability of 99.94%. The results differ with the logarithmic distribution of number length. E.g. with mu=11 digits and sigma=2 digits, the probability of 4 digits is 85%. With mu=12 digits and sigma=3 digits, the probability of 4 digits is 98.3%. With mu=7.5 digits and sigma=4 digits the probability of 4 digits is 99.95%. With mu=12 digits and sigma=2 digits, the probability of 4 digits is 47.64%. Sebastian --126.96.36.199 11:07, 3 February 2013 (UTC)
The 11:59 subtle joke is slightly reinforced as the countdown steps over 2400. Sebastian --188.8.131.52 11:11, 3 February 2013 (UTC)
Could "the odds are in our favour" be a reference to the hunger games? 184.108.40.206 (talk) (please sign your comments with ~~~~)
- If you had read all the comments, you would have seen that someone else already thought the same, and nesting your comment below his/hers would make more sense. But that's just me grammar naziing around. 220.127.116.11 00:05, 16 February 2014 (UTC)