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?
Best Answer
The probability of not getting a head on the first toss is $\frac{1}{2}$, the probability of not getting a head on the second toss is $\frac{3}{4}$, the probability of not getting a head on the third toss is $\frac{7}{8}$, and so on.
So the probability of not getting a head on the first $n$ tosses is $$\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{7}{8}\cdots \cdot \frac{2^n-1}{2^n}.$$ We take the limit as $n\to \infty$. The convergence is rapid. The infinite product is about $0.289$, a bit under.