If my very ropey understanding is correct, 10 repetitions of something with 2 possible outcomes has 1024 possible combinations. So if we ignore the green(s) for the sake of simplicity, the chance of getting 10 consecutive reds is 1/1024th the chance of not getting 10 consecutive reds.
almost - it isn't '1/1024th the chance of not getting 10 consecutive reds.'
it is simply 1/1024
and the chance of it not occurring in those 10 reparations is therefore 1023/1024
if you want to describe it as in the above sentence then it is 1/1023 the chance of not getting 10 consecutive reds... in other words the event that it doesn't occur is 1023 times more likely
or, in gambling terms, the odds of it occurring are 1023:1 against