Saying something happens with probability 1 does not mean it always happens; saying something happens with probability 0 does not mean it never happens |

Rather, suppose the process is repeated again and again. We compute the fraction |

If the fraction goes to 1 as the total number of repetitions goes to infinity, the event happens with probability 1. |

If the fraction goes to 0 as the total number of repetitions goes to infinity, the event happens with probability 0. |

Under forced circumstances, this perfectly sensible definition can lead to some counterintuitive results. |

For example, suppose a coin is tossed infinitely many times, and heads comes up on only the |

1^{st}, 2^{nd},
4^{th}, 8^{th}, 16^{th}, 32^{nd}, 64^{th},
128^{th}, 256^{th}, 512^{th}, 1024^{th}, ... |

tosses. |

That is, heads comes up only on the tosses numbered by powers of 2, and all other tosses give tails. |

Here we plot the fraction (vertically, the top of the vertical line is at 1) vs n for n to 1024. |

The fraction goes to 0, so we are left with the odd situation that heads comes up infinitely many times, yet the probability of getting heads is 0. |

So: probability = 0 does not mean an event never occurs, just that it is increasingly unlikely with more repetitions. |

Also, probability = 1 does not mean an event always occurs, just that is is increasingly likely with more repetitions. |

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