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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?

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    "A sequence of 100 consecutive heads will always occur infinitely many times" ... uh... Why? The sequence 01010101.... has zero ocurrences2011-08-23
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    +1 to leonboy's comment. Remember that any particular sequence of 0's and 1's is "possible" in some respect. Each individual sequence has probability 0. So, it isn't that it is "impossible" to see a sequence like 01010..., it's just that the set of all outcomes such that stretches of 100 consecutive heads don't occur infinitely often has probability 0.2011-08-23
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    @leonbloy: That's not random...2011-08-23
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    @Guffa: Give me a sequence, I'll tell you it's not random, therefore no sequence is random, therefore this problem makes no sense. That's the road down which your logic goes.2011-08-23
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    @Guffa: technically, to say that a particular outcome of a random process "is (or is not) random" does not make sense. See anon's answer.2011-08-23
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    @anon: Any sequence that you construct with the specific goal of not containing a certain sequence, will not be random.2011-08-23
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    @guy, your advice is to *Remember that any particular sequence of 0's and 1's is "possible" in some respect.* In fact I would rather say that any particular sequence of zeroes and ones is impossible... :-)2011-08-23
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    @Didier, I put "possible" in quotation marks and added the qualifier "in some respect" precisely because I didn't want to get into this :) That was the best I could do to try to get the general idea across without being formal.2011-08-23
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    @guy, I know, I know, but I could not resist the pun... As the saying goes, *l'occasion était trop belle.*2011-08-23

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