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When heads has probability p the probability for any n-length sequence of all heads is p^n. For p between 0 and 1 this vanishes as n goes to infinity. The reas
by conjectures 10y ago
When heads has probability p the probability for any n-length sequence of all heads is p^n. For p between 0 and 1 this vanishes as n goes to infinity.
The reason is that the conjunction of some events is always less (than or equal to) probable than the events themselves. This should make intuitive sense as the middle bit of a Venn diagram. Even more so for a conjunction of events with a conjunction of events. When taken to the infinite limit you tend to end up with the conjunction having probability 0 or 1.
- danbruc 10y agoI have no problem with that, all infinite sequences of coin tosses have probability zero. The question is whether a infinite sequence of all heads is an obtainable outcome which would be an counterexample to the statement that the relative frequency converges to 0.5 in the limit.