Let's say I start off with a 50/50 chance at winning the lottery. But I lose.
Now my chance is only half as good, or 25%. I lose again.
Now the chance is 12.5%. Same result.
If this continues all the way down to very low numbers, will I EVER in an infinite amount of time win the lottery?
What if every time I received a ticket, I had a million times less chance of winning than I had before?
On the first flip, you have a $\frac12$ chance of winning.
Assuming you lost (with probability $\frac12$) you win with probability $\frac14$, which gives you an additional $\frac18$ probability of winning, for a total of $\frac58$.
Assuming you lost again (probability $\frac38$) you get another shot at probability $\frac18$, for an additional $\frac3{64}$, for a total of $\frac{43}{64}$.
Continuing this on, the probability $p_k$ of having won after $k$ flips is given by $$ p_0 = 0\qquad p_{n} = p_{n-1} + 2^{-n}(1-p_{n-1}). $$ Notice that, except from the first step where you add exactly probability $\frac12$, at each step $k$ you add strictly less than $2^{-k}$. Because $$ \sum_{k = 1}^\infty 2^{-k} = 1, $$ it is possible with positive probability for you to lose the lottery (although your chances are still pretty good).