# What combination of even and odd numbers would you prefer for $5$ digit lottery ticket?

I sometimes play lottery. The tickets are $5$ digit numbers, for example $34298$. I always buy a ticket which have $2$ even, $3$ odd numbers on it or vice versa, even though I know that it does not make sense mathematically (or does it?). I would never choose a ticket like $13579$ or $11335$.

What would you prefer?

To make an analogy, imagine that there are many black and white balls in a bag. If you take $5$ balls from the bag it is more likely that you would get some black and some white balls. Similarly if you wrote even numbers on black balls and odd numbers on white balls, more likely you would get some even and some odd numbers.

Is my analogy wrong?

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Your analogy, as you have stated it, is correct. It is more likely to draw, say, 2 evens and 3 odds (in some order) than it is to draw 0 evens and 5 odds.

I think Cameron's point is that the analogy is irrelevant to your chances of winning because the amount of evens and odds on your ticket does not by itself determine whether you win, but rather the particular numbers determine this. (That is, the even numbers are not all indistinguishable from one another for the purpose of determining whether your ticket wins.)

Where you go wrong (I think) is in the next step of your reasoning, which you have not stated explicitly. The fact that "2 evens and 3 odds" is more likely than "0 evens and 5 odds" does not mean that $1,2,5,6,9$ (for example) is more likely than $1,3,5,7,9$—these two sequences have the exact same chance of being drawn.

The reason "2 evens and 3 odds" is more likely than "0 evens and 5 odds" is not because any particular sequence fitting this description, e.g. $1,2,5,6,9$, is more likely than any other particular sequence, but rather because there are more sequences fitting this description than there are sequences fitting the description "0 evens and 5 odds."

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Your analogy is incorrect, as the black balls (for example) are indistinguishable from each other, but the even-numbered balls (for example) in a lottery are not. It's worth noting that $11335$ is still an objectively terrible choice for a lottery ticket, though, since repeats will not happen.

As for my personal preference, I'd invest or save my money.

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