I was reading about Kolmogorov's zero-one law specifying:
a certain type of event, called a tail event, will either almost
surely happen or almost surely not happen
I came to this example:
In an infinite sequence of coin-tosses, a sequence of 100 consecutive
heads occurring infinitely many times is a tail event.
That can't be true, can it?
In an infinite sequence of coin-tosses, any specific sequence will occur infinite times. A sequence of 100 consecutive heads will always occur infinitely many times, not almost surely.
Saying that a sequence will occur any less than infinite many times actually get absurd. If the 100 consecutive heads occur any finite number of times, if I then get 99 consecutive heads any time after that, the next toss will not be random, but it has to turn up tails.
So, am I missing something fundamental?
Best Answer
There are infinite sequences of coin flips that do not contain a single stretch of 100 consecutive heads. In fact, there are uncountably infinitely many: let $t_1,t_2,t_3,\dots$ be an infinite sequence of numbers from $\mathbb{N}$, and $h_1,h_2,h_3,\dots$ be an infinite sequence of numbers from $\{0,1,2,\dots,99\}$. Then a sequence of $t_1$ tails, $h_1$ heads, $t_2$ tails, $h_2$ heads, and so on is a sequence with no stretch of 100 heads.
When dealing with infinity, "almost surely" deals with situations that occur with probability $1$ according to the formal notions associated with a probability space. This does not imply there are no valid situations where an event occurs or a hypothesis is sasisfied, as Wikipedia says: