Should I put number combinations like 1111111 onto my lottery ticket?
math.stackexchange.com
math.stackexchange.com
So I can answer the OP: No, you shouldn't, because you'll really piss off your dad.
When you pick random numbers, you might think that those look like the numbers that are usually drawn, and "almost every time, somebody wins so why not me?" etc. But then imagine that you chose 11111111. The idea that this is the combination that will be drawn is laughable. You couldn't possibly imagine it. It would be on the news, everybody would talk about it. Well, you have those same odds with your random numbers!
It's impossible actually with the lotteries I'm familiar with because no number can be chosen more than one time.
The stat that did it for me was when I heard that I'm far more likely to be murdered than win the lottery
I personally like "You're more likely to die redeeming the prize than you are to win the lottery itself."
The logic is, sharing the prize greatly hurts the expected value (when N people win with the same numbers, they split the jackpot 1/N). Certain people like to play numbers corresponding to their birthday, anniversary, or other meaningful dates that range from 1 to 31. In the big US lotteries (Mega Millions and Powerball), the numbers range as high as 59, so picking a bunch of numbers in the 40s and 50s greatly reduces the chance of having to share the prize.
Another interesting story I've heard (no source sadly): one time the winning numbers were exactly the same as the "lucky numbers" printed on the back of a fortune cookie. An inordinate number of winners shared the prize, all Chinese food eaters who received the same fortune that week.
There was a set of numbers that appeared in one of the episodes for the TV show "Lost" and coincidentally, 4 of the numbers matched the numbers from the MegaMillions jackpot. Exactly 41,763 people matched those four numbers and each came off with $150 apiece.
source: http://articles.latimes.com/2011/jan/06/entertainment/la-et-...
That makes me wonder what's the best meta-algorithm to use when deciding on algorithms to use. Any good way to improve your number generating process is likely to be used by other people, ipso facto. And even if you try to derive a process that takes that into account, the more powerful your process is, the fewer the number of people it would take to moot your strategy.
And yet some processes--pulling numbers off of a fortune cookie--are genuinely horrible. Which seems like a mind-bendy paradox to me. There's this odd space between using a simple random number generator and using some kind of smart algorithm to do it that seems good. But the very fact that an algorithm exists to choose sone set of numbers seems to negate its value, so it seems almost beyond us to beat a random number generator.
Briefly, on the first anniversary of 9/11, 911 was a winning three-digit lottery number in New York. However, 5631 people chose that number so the payout was comparatively small.
I'm reminded of this incident in bulgaria where the lottery came up with the same numbers two weeks in a row
Notice how EIGHTEEN people had to share the pot (presumably) because they thought they were being clever in picking numbers noone else would pick.
I was so damn excited for about 3 minutes until I compared the dates on the lottery site.
Eg. numbers 1-12 and 1-31 are more often used as people play significant dates, leading to high numbers having higher ev (still negative though).
I remember reading a paper somewhere on this but can't for the life of me source it.
There are still 5 separate "1" balls, so still 120 ways to draw 5 1's in a row.
EDIT: Never mind, I get it. Removing the first "1" reduces the chances of getting the second "1", etc.
That is a point, but it's not the point maxjus was making. Two different issues:
First, whether order matters - whether a drawing of "17, 23, 31" is the same as a drawing of "23, 31, 17".
Second, whether balls are replaced - whether you put back the "17" ball after it's been drawn, before you pick the next number.
Assuming balls numbered 0 through 49:
- If order matters and balls are replaced, then "1, 1, 1" is as likely as "1, 17, 23".
- If order doesn't matter and balls are replaced, then it depends on how many of each number there are. If there's one of each number, "1, 1, 1" is six times less likely than "1, 17, 23", as there' six different ways to make the latter (since "1, 17, 23" and "23, 17, 1" are the same), but only one to make the former. If there's three balls of each number, they're equally likely.
- If order matters and balls are not replaced, then it depends on how many balls of each number there are. If there's one of each number, "1, 1, 1" is impossible. With three of each number, "1, 1, 1" is still less likely than "1, 17, 23", as, e.g. for the second number, there's two "1"s but three "17"s. (More specifically, there's six ways to draw "1, 1, 1" (3x2x1), but 27 ways to draw "1, 17, 23" (3x3x3)).
- If order doesn't matter and balls are not replaced, then it depends on how many balls of each number there are. If there's one of each number, "1, 1, 1" is impossible. If there's three of each number, "1, 1, 1" is way less likely than "1, 17, 23" - there's six ways to draw "1, 1, 1", but 162 ways to draw "1, 17, 23" (9x6x3).
Edit: posted the above as an answer on stackexchange, since all answered posted so far have silently assumed that order matters and balls are replaced(!).
The human mind, taken as an algorithm, might prefer short programs that are fast to execute and require little energy to run.
Vitanyi proved that highly random strings must include repetition and patterns. Very crudely (and because I don't quite get all the maths):
A string-generating program is more random when its output is less predictable. If you don't have to worry about predicting patterns and repetitions, this makes the rest of the program output more predictable. Thus, highly random generators should output patterns and repetitions too.
So, if we take the lottery numbers to be randomly generated (this is important), we must expect patterns like 1111111111 too.
We can approach Kolmogorov complexity (not computable) through compression. Basic Run-Length Encoding (RLE) of 1111111111 gives us (10,1). This is a very short and simple program and it seems that humans prefer such programs over more complex ones.
If we don't want to share the lottery with people who look for patterns in previous draws, we should generate a string, that, when added to the previous draws and compressed produces the longest output.
Let's say the previous two draws were 1111111111 and 22222222222, or compressed with RLE (10,1),(10,2). Humans who will optimize for patterns will produce strings that result in a shorter size. Let's say some humans play the numbers of latest draw (222222222). That will result in a compressed string of all 3 draws: (10,1),(20,2). This has the exact same length as the compressed string of only 2 draws. The previous draws and the prediction share patterns that allow the compressor to produce smaller strings.
Optimizing for maximum complexity would have you pick a draw like "2453160798" (not possible to compress with RLE). A string like "1234567890" also does not compress with RLE, but it easy to spot the simple, short program generating it (n+1..).
What happens when some humans increasingly pick numbers that appear random, but in fact are very short programs or carry cultural relevance (a range starting at _n_th decimal of Pi, validation key of pirated windows XP, dates of birth of my first two children, my lucky 100-digit number).
If our compression algorithms were perfect they would be able to generate the shortest program that describes another, bigger (or equal size) string. But compression is not perfect. Most compression algo's would struggle to compress a range of decimals of pi at n, as something smaller than a completely random number string. There is a simple pattern, but it won't be found in a reasonable time.
Luckily we can use search engine indexes as our compression algorithm! If a string like "12345" has 1 million hits and string "67890" has 1 million hits and both strings together have 500k hits, we can say that these strings often co-occur (probably in the form 1234567890).
If a certain date carries cultural significance (1970-01-01), it would have more hits/results, than a date that carries zero, to little, significance (2641-10-02). If 8 is your lucky number, chances are (Chinese) people local to you, also have picked this as their lucky number.
Then the key to picking a less predictable sequence is to pick a sequence that doesn't appear (a lot) in the search engine index. With a little luck we don't even have to get the hits for previous draws, as a large search engine index is likely to already contain these draws.
So if we want to optimize for not sharing with a large amount of people, and we assume that the lottery numbers are highly random: pick number ranges that do not compress well with the previous draws and pick number ranges that have a low result count in a search engine like Google.
[1] Tasked to pick a random number between 1 and 4, there is a bias to picking 3. When between 1 and 10, there is a bias to picking 7. Why this is? 3 and 7 may appear more special than 6 (23) or 4 (22). Also the phrasing of picking _between_ 1 and 4 might throw of humans to NOT pick 1 or 4. A randomrange program doesn't care for such human subtleties.
Random passwords are harder to remember for humans (this might be because the "mind program" to generate them is complex and requires more energy). Human passwords often contain dictionary words and commonly used patterns (which are faster and cheaper to regenerate).
Whenever theres a big draw here I just buy 1 ticket of 123456. I really want it to win one day and watch the idiots come out of the woodworks and call it rigged. You would be surprised how many relatively smart people think that such numbers are more unlikely.
Its also funny to hand 35 cents over at the newsagents while everyone use is dropping at least $5 on the draw
(odds of winning anything = the same. likelihood of having to share the jackpot = smaller.)
Given that I've heard people repeating this advice for probably at least 15 years, I imagine that 123456 is probably among the most popular combinations played.
I suspect that numbers with pattens are more commonly picked and like someone else said the numbers inline with birthdays
Best is probably a real random number generator.
To add to the vig, the Mafia would not let you bet on numbers like 100, 111 and such, because they didn't want to have a big payout day. Of course, if any of those numbers came up they wouldn't pay anybody out.
For example normally, if the numbers chosen are truly random, the odds of 1111111 being the winning number is 1 / 10 000 000.
But lets say in the powerball bin, there is 7 '1' balls, 7 '2' balls and so forth, then the odds that first drawn number will be '1' is 7 / 63. That ball is then removed from the bin. The odds that second number will be '1' is now 6 / 62 etc.
That brings the odds of 1111111 to 7x6x5x4x3x2x1 / 63x62x61x60x59x58x57 = 2 / 1 000 000 000.
Quite a bit less... The effect will vary depending on how many balls of each number there is in the bin, and how balls are replaced once selected. Some powerball lotteries has a preselection draw which determines which balls are in the final draw, but the same argument is still valid.
> heads-tails-heads-tails-tails-heads
Now what are the chances that you're going to flip:
> heads-heads-heads-heads-heads-heads
Is that chance equal to you flipping the winning combination?
And I don't mean for this to happen anywhere between the first and the million'th try, but the first try.
Sequences have different entropy values, which means they are not equally random. http://en.wikipedia.org/wiki/Entropy_(information_theory)
Edit:
Also (but unrelated to the above), can you look at it this way?
There are X amount of same-number sequences (ex: 2222222, 3333333).
There are Y amount of non-same-number sequences (ex: 2449776, 4553219).
Y >> X
If that is true, then getting a same-number sequence is not just as likely as getting a non-same-number sequence because if the lotto draws from a pool of all sequence numbers, there are much much more non-same-number sequences in there.
If that is wrong, I'd love to know why. Can someone summarize why entropy would make those sequences unequally random for a fair coin? I guess it's not clear to me why they'd have 'different entropy values'.
There is one combination with 6 Heads.
There are twenty combinations with 3 Heads and 3 Tails - therefore 3 Heads and 3 Tails is twenty times more likely to come up, true.
However, the specific sequence HTHTTH itself is no more likely than HHHHHH. It's just that a sequence with the same distibution of H and T is more likely.
If HHTHTT comes up and you picked HTHTTH, it might look similar but it's no good.
==
It's more likely that a lottery will result in a non-same-number sequences than a same-number sequence: true. This is because they are a much! bigger group of numbers.
However, an individual member of the same-number group has the same chance as an individual member of the non-same-number group.
2222222 is just as likely to come up as 2449776 is just as likely to come up as 9667224.
The Quebec lottery published their most popular numbers a few years ago [1]. They were:
7-14-21-28-35-42
1-2-3-4-5-6
4-8-15-16-23-42 (the numbers from the tv show Lost)
And, while I haven't confirmed it, I suspect that in games like Powerball, where winning tickets are determined by two independent sets of balls, people stick with their habit of choosing numbers in numerical order and not repeating numbers, so 4-12-17-24-36 (42) would be a more common pick than 4-12-17-24-36 (24).
[1] http://lotoquebec.com/loteries/nav/en/useful-information/pop...
But forgetting the actual fair drawing process and just focussing on randomly choosing a number from a set of numbers...if you divide a set of numbers into significant-looking and insignificant-looking, would it not be true to say that a random choice is more likely to look insignificant and random, than significant and ordered, purely because there are far more insignificant looking ones?
(I'm not saying that 11111111 isn't equally likely to be drawn in a lotto here)
Now the 50% is ofcourse a little joke. But there is some truth in it. Randomness doesn't mean the numbers change all the time. The probability that the numbers change is ofcourse very very high. But random doesn't equal change.
http://txlottery.org/export/sites/lottery/Games/Lotto_Texas/...