Lottery probability

The chances of the next number being between 1 and 10 is $\frac{7}{49}$, as opposed to the probability of it being between 11 and 20 being $\frac{10}{49}$. So, among other things, it is less likely that a lottery ticket will have only numbers between 1 and 10, as opposed to numbers between 1 and 20. However, that does not mean that a given ticket with numbers between 1 and 10 is less likely then a given ticket with numbers between 1 and 20. The fact that there are more tickets with numbers between 1 and 20 exactly cancels with the higher chance of getting such a ticket.

Example: You have a bag of pens; a red pen, and 99 blue pens numbered 1-99. The chance of you getting a red pen is only .01 whereas the chance of oyu getting a blue pen is .99, but the chance of getting any given blue pen (say, #42) is .99 / 99 = .01, which is the same as the chance of getting the red pen.

Having chosen 1,2,3 all possibilities (there are 46*45*44/6) that include 1,2,3 are equally likely and all possibilities that do not include 1,2,3 have probability zero. The psychological error comes from grouping "numbers less than 10" together. As Moron commented you could also group "numbers containing a digit 2". Each of these are legal groupings and the probability of selecting all numbers from within them can be calculated. But it is still true that any particular combination is equally likely at the outset. A draw of 13,18,29,33,36,45 looks "random" to our eye and we group it with all other "random looking" combinations. From this point of view, a "random looking" combination is highly likely.