

Should I put number combinations like 1111111 onto my lottery ticket? - jontro
http://math.stackexchange.com/questions/467575/should-i-put-number-combinations-like-1111111-onto-my-lottery-ticket

======
crux
When I was thirteen or fourteen, my father—who was pretty well-off but played
the lotto very regularly—let me fill in one of his tickets for a bit of fun.
I, being the precocious young person that I was, entered 1,2,3,4,5,6,
reasoning that since the lotto was random, any given combination was equally
likely to come up. Pleased with myself and with the laws of mathematics, I
showed it to my dad, who immediately gave me a massive bollocking-out for
having wasted a ticket on a number that was sure never to come up.

So I can answer the OP: No, you shouldn't, because you'll really piss off your
dad.

------
kristiandupont
This is actually one of the better ways of illustrating how low the
probability of winning is.

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!

~~~
barking
"The idea that this is the combination that will be drawn is laughable"

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

~~~
kabouseng
I live in Africa, so my odds of winning the lottery isn't too bad...

~~~
Ellipsis753
That just makes you more likely to be murdered. You're still going to be "far
more likely" to be murdered than to win the lottery.

------
leelin
I've heard picking a lot of high numbers is a better strategy than picking
pseudo-random numbers.

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.

~~~
jmharvey
Picking _some_ high numbers is good, but you have to be careful about picking
_all_ high numbers, since many other lottery players have also heard of this
strategy. In a 6/46 game, of the 9.3 million combinations, only 5005 consist
entirely of numbers higher than 31, while 736,281 are entirely "birthday"
numbers. So if for every 150 people who play birthday numbers, there's one
person who thinks they're outsmarting everyone and plays all high numbers,
they're not actually improving their chances of picking an unpopular number.

~~~
scarmig
Ideally you want to pick the set of numbers that's least likely to be shared
with other players. However, the data used to derive any algorithm you use to
determine that set of numbers will be publicly available.

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.

~~~
sopooneo
I bet choosing randomly is going to come very close to being ideal. The amount
of numbers that other people can buy up will be an insignificant.

------
Lewton
[http://www.sbs.com.au/news/article/1093277/Identical-
lottery...](http://www.sbs.com.au/news/article/1093277/Identical-lottery-
numbers-drawn-twice-in-a-row)

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.

~~~
DoggettCK
My wife picked the previous drawing's numbers once. Once.

I was so damn excited for about 3 minutes until I compared the dates on the
lottery site.

------
thret
People who think all lottery numbers have equal value are mistaken. Division
payouts are divided among the number of winners, so combinations that are less
frequently used have a higher expected value.

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.

------
maxjus
It should be noted that for the actual lottery, the advice given is wrong.
Because in many real lotteries the first five digits can be in any order, 1 1
1 1 1 would be less likely than, say, 13 23 42 33 2, because, though there is
only one way to end up with 1 1 1 1 1, there are 5! == 120 ways to achieve the
latter (I am ignoring the "bonus ball").

~~~
jstanley
I think that's incorrect. If you secretly label the "1" balls with secondary
numbers, do the odds change?

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.

~~~
SEMW
> 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(!).

------
alxbrun
I'm surprised no answer mentions the Kolmogorov complexity [1]. There's a
fascinating link between how a given series seems "probable" to humans and its
Kolmogorov complexity.

[1]
[http://en.wikipedia.org/wiki/Kolmogorov_complexity](http://en.wikipedia.org/wiki/Kolmogorov_complexity)

~~~
blauwbilgorgel
Humans are notoriously bad at generating random numbers. [1] Very random
strings are described by very complex programs.

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 (2 _3) or 4 (2_ 2). 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).

------
jasey
I dont play the lotto because the odds are terrible and I dont like the "OMG
my life is awful but would be awesome if only I got lucky and hit the jackpot"
attitude.

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

~~~
dagw
_Whenever theres a big draw here I just buy 1 ticket of 123456._

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.

~~~
jasey
Its hard to know for sure, my guess is your right but its hard to know without
the data.

I suspect that numbers with pattens are more commonly picked and like someone
else said the numbers inline with birthdays

------
PaulHoule
Back in the 1970's, the Mafia ran a numbers racket in NYC (and Chicago and
presumably other cities) that was very similar to the "Pick 3" game run by the
New York Lottery. That is, you would pick a 3 digit number and if it matched
the last three digits of the balance of the U.S. Treasury that day.

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.

------
kabouseng
Another point to consider is how is the winning number selected. Here the
lottery numbers is decided through a powerball mechanism
([http://en.wikipedia.org/wiki/Powerball](http://en.wikipedia.org/wiki/Powerball)).
So given that there is only a fixed amount of numbered balls the winning
number is drawn from, this skews the probability of the winning number being
111111 versus another random number.

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.

------
elmuchoprez
Since lottery jackpots are traditionally shared among everyone who picked the
winning combination, the only skill/logic in picking a number is picking one
that you believe is an uncommon choice. That way, should you win, there's less
people to share it with. And there probably is statistical evidence to back up
the the idea that some numbers are chosen by people more often than others
(ie: somehow everyone's lucky number is 7).

------
powertower
Take a coin. The winning flip combination will be a somewhat random sequence
like:

> 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)](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.

~~~
zipppy
My answer is 'yes', that those two are equal. And that it doesn't matter if
the first try is specified -- it is an equal chance for either of those to
occur on the first try.

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'.

------
jmharvey
Of course, 1111111 has as good a chance of winning as any other number, but
since most lotteries are parimutuel (i.e. the prize is divided evenly among
all winners), it makes sense to try and pick a number that no one else is
going to pick.

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)
    

Also, from personal experience, it seems that more people play "birthday"
numbers (numbers 1-31) than higher numbers.

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...](http://lotoquebec.com/loteries/nav/en/useful-
information/popular-selections/accueil)

~~~
sopooneo
And yet I bet no winner of a divided prize wishes they had picked a less
popular number.

------
anjc
Stupid question: the reason he's questioning the equal probability of 1111111
being drawn is because it _seems_ significant. It looks like it is less likely
to be drawn. But we know it is equally likely.

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)

------
teahat
I remember reading several years ago that in the UK lottery 10000 people every
week play 123456. And thus would be very disappointed at their £500 payout.

------
ohwp
Yes you can use 1111111. The chance that you win is 50%, you do or you don't.

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.

------
dllthomas
No; you shouldn't put any numbers onto your lottery ticket because you
shouldn't have a lottery ticket.

------
skc
For me, the easiest way to visualise that you have equal probability between
the two sets of numbers is to simply visualise the balls, but without the
numbers. Now, what are the odds of drawing blank set of balls A. Then, what
are the odds of drawing blank set of balls B.

------
moron4hire
This needs to be clarified, because there are some games that play out of a
single bin for all numbers and some games that play out of separate bins for
each number. In the case of a single bin, a series like 111111111 is less
likely to appear than a series like 123456789. The reason is that there are a
finite number of balls in the bin. If there 9 count of each number of ball,
then the probability of each ball i (from 0 to 8) coming up X in turn is (9 -
i) : (81 - i), whereas any other number, in its turn, would be 9 : (81 - i).

~~~
astrodust
Which lotteries use single bin? That seems to have a non-uniform distribution.
All of the ones I know use one bin per digit.

~~~
whatusername
Pretty sure that the major lotteries in Australia are single bin.

------
fnordfnordfnord
Has anyone ever analyzed the distribution of past lottery numbers for
randomness?

~~~
stonemetal
On the Texas lottery site they list counts of how often every number is
picked. It wouldn't surprise me if other state lotteries were similar.

~~~
fnordfnordfnord
Correct, you are.

[http://txlottery.org/export/sites/lottery/Games/Lotto_Texas/...](http://txlottery.org/export/sites/lottery/Games/Lotto_Texas/Winning_Numbers/lottotexas.csv)

~~~
stefap2
there is a sudden drop in frequency of numbers above 48

~~~
stonemetal
Numbers above 48 were added after the program had run for a few years. I am
not sure if they changed providers or just the program but they went from a
straight pick 6 to a powerball setup with a different number range. It is easy
to see they didn't reset the stats when they made the change because the "mega
ball" count is much lower than the regular number count.

