Conceptual question on Probability today I've stumbled in a interesting question about probabilities while talking to a friend, I don't know whether here is the best place to ask, but as it involves probabilities, I'll try.
Given a set of numbers $1,\ldots,N$ as a sample space in a lottery ticket, we have to mark a subset of $M$ numbers.
Why do we have a common sense that we have better chances choosing a sparse subset of numbers instead of a contiguous subset. Isn't the probability the same, or is there deeper math involved that I'm not aware of?
Thanks in advance.  
 A: The probability of any number combination being drawn is exactly the same, assuming the lottery is fair (which it is virtually guaranteed to be). However, if you want to maximize your expected winnings, then you have to take into account how many people you split the prize with. If you choose 1,2,3,4,5,6 probably a lot of other people will choose this too so if you win you have to split the prize many ways so your expected winnings are significantly lower. So you should either choose randomly or try to figure out what number combinations are the least likely to be picked by other people.
A: While it is true that all subsets of size $M$ have the same probability,
the intuition that some subsets are 'rare', or 'suspicious' is not
entirely foolish.
Partly, this has to do with the ever-present doubt whether the lottery
is really fair. What patterns would provide evidence of unfairness.
Suppose we wonder whether the $N$ numbered balls are properly mixed
before drawing. One possible consequence of improper mixing would
be for the numbers drawn all to be low numbers, say all below $N/4$,
or all of them to be high numbers, say above $3N/4$. 
If we keep track of the order in which the numbers are drawn, we
could do a runs test for randomness. A very long run of numbers lower
than $N/2$ anywhere among the $M$ draws would cause a sample to 'fail' the runs test and
lead to the conclusion that the drawing mechanism isn't a random
one.
[Note: Suspicions have not always been unwarranted. Google 'unfair
draft lottery' to find information on pretty clearly biased
US draft lotteries during the Viet Nam War era. Before the method
of 'randomization' was refined, it seems that far too many December
birthdays were chosen early.]
Hypothesis testing often depends on picking outcomes that look
extreme assuming the null hypothesis to be false in a particular way. A simple non-lottery example
would be a test of whether a die is fair or biased in favor of 6's.
Suppose we get all 6's in ten rolls of the die. That outcome has the
same chance as any of the other $6^{10} = 60,466,176$ possible (ordered)
sequences. But, because this extreme outcome is especially likely for
a die biased in favor of 6's, no sensible person would believe the
die is fair. 
Many--maybe most--gamblers enter a game with hopes (or suspicions) that
the allegedly random process will somehow be biased for them (or against).
Given such mind-sets, it is hard for them to believe that all of the
outcomes are really likely.
A: While Bruce's explanation of possible unfairness is plausible, I think our intuition is rooted in something else, which we might think of as the fungibility of a given set of six numbers.
Suppose the lottery selection is six numbers from the set $\{1, 2, \ldots, 50\}$.  The subset $\{1, 2, 3, 4, 5, 6\}$, though no less likely in fact than $\{3, 17, 20, 31, 44, 48\}$, is nonetheless perceived to be less likely, because the probability that something "like" $\{1, 2, 3, 4, 5, 6\}$ (for some not altogether well-defined sense of "like") is less likely than something "like" $\{3, 17, 21, 30, 44, 48\}$.  Probably the only thing "like" $\{1, 2, 3, 4, 5, 6\}$ in most people's minds is itself, whereas many subsets are "like" $\{3, 17, 20, 34, 41, 48\}$.
For instance, I wonder how many readers noticed that I made small changes in the more random-looking set, each time I referred to it.
This notion of fungibility is related to information-theoretical notions of entropy and Komolgorov complexity, and those notions might be used to express the fungibility, but I'm not sure that human psychology can really be accurately reduced to those notions.  It's probably best, in my opinion, to leave it vague and think of it as a general notion of fungibility.
