TITLE: What happens if I toss a coin with decreasing probability to get a head?
QUESTION [5 upvotes]: Yesterday night, while I was trying to sleep, I found myself stuck with a simple statistics problem. Let's imagine we have a "magical coin", which is completely identical to a normal coin but for a thing: every time you toss the coin, the probability to get a head halves. So at t = 1 we have 50:50, then 25:75, then 12.5:87.5 and so on. At t = ∞ we are going to have 0:100.
My question is: if I toss this magical coin infinite times, can I say I am sure I am going to get at least one head? On one hand, I thought, the law of large numbers states that if an event is repeated infinite times, every state that is possible is bound to happen. On the other side, however, at t = ∞ the probability to get a head is zero.
Surely the solution of the problem is fairly easy, so what am I doing wrong?

REPLY [3 votes]: The probability of getting all tails is
$$
\prod_{k=1}^\infty\left(1-\frac1{2^k}\right)\tag{1}
$$
For $0\le x\le\frac12$, we have that
$$
\log(1-x)\ge-2\log(2)\,x\tag{2}
$$
Therefore, since
$$
\sum_{k=1}^\infty\frac1{2^k}=1\lt\infty\tag{3}
$$
we have
$$
\prod_{k=1}^\infty\left(1-\frac1{2^k}\right)\ge\frac14\tag{4}
$$
Thus, the probability of getting a head is less than $\frac34$.