Please forgive me if this is too outrageously vague a question; it is only meant as idle speculation, really, but sometimes it is good to let one's mind wander a bit.
In the last year or so there has been a huge increase in advertising for gambling, and apart from the ethical and psychological aspects of gambling, I thought about what one can compare one's chances of actually winning in, eg. a lottery. So, my question is, how does it compare to finding a needle in a haystack? The relevance of this question is that when people buy a lottery ticket, they don't really understand how unlikely winning is, but most have an intuition about the chances of finding the nail in a haystack.
So, if we consider a realistic size of haystack containing one needle, and I were to put my hand somewhere in the haystack and pull out whatever I grabbed hold of, what are the chances that it might be the needle - and how does it compare to winning in the lottery?
I haven't touched combinatorics or probability theory since I left university some 35 years ago, so I have very little skill in this area, but I realise it may be unanswerable, so perhaps this question is as much about the method - starting with deciding which lottery (eg. the EuroLotto, or whatever it is called) and which haystack, as well as deciding how much freedom one has in placing the hand.
Just as a toy example, the chance of a winning ticket for the UK National Lottery is around one in $45$ million. Let's find out how big a cubic haystack would have to be for me to have the same odds of finding a needle in it.
Say I can grab a handful of radius $r$ and the haystack is a cube of side $s$. Then $$s^3 = 4.5\times 10^7\times\frac{4\pi}{3}r^3$$
Taking the cube root of both sides, $$s \approx 573r$$
Even taking a fairly small $r$, say $r\approx 5\text{cm}$, this means a haystack of side around $28\text{m}$, which is very big.
Obviously these calculations involve approximations and assumptions, but hopefully the above shows how you could get a rough idea (which I think is appropriate here; the haystack is considerably larger than a real one, but isn't, say, the size of the moon).