Tossing and turning

Which is more likely in tossing a fair coin ten times, that you alternately get head, tail, head, tail and so forth, or that you get ten heads in a row? Let us calculate. For the alternating series we have half a chance of getting a head and then half a chance of getting a tail and then half a chance of getting a head and so forth. So the overall probability of this series is (½)10 = ½10 = 1/1024. On the other hand, the chance of getting ten heads in a row is one-half times a half times a half, and so forth, and equals (½)10 = ½10 = 1/1024. In other words, the two events are equally likely, or, as it turns out, unlikely.

Did you think that the sequence corresponding to head tail head and so forth would be more likely? Perhaps you were thinking of a related problem? Which is more likely in tossing a fair coin ten times, that you get five heads and five tails in any order or that you get ten heads in a row? Here the point is that there are many more sequences that satisfy the condition of five heads and five tails than the alternating one originally defined.