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Let's say I've coin where P(head on first toss) = P(tail on first toss) = 0.5, and P(head | previous toss was a head) = 0.8 and P(tail | previous toss was a tail) = 0.8.

Suppose I have a given realization of these coin tosses, say H, H, H, H, H, H, T, T, T. It is clear that there is serial correlation, since the first six tosses were heads and the next three tosses were tails. In other words, given that a toss was a head, I can predict whether or not the next toss was a head or tail better than chance.

If I were to randomly rearrange the order of the flips for a given realization, will there still be serial correlation (on average)?

My intuition is that there won't be serial correlation due to the randomness, but I'm not sure how to prove it.

wwl
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  • But what if you would do it with e.g. 0.99 instead of 0.8? You will get most probably all heads or all tails. Rearranging is without any effect then. It might gradually return though by increasing the total number of tosses. – drhab Oct 11 '17 at 13:22
  • I'm not asking about the number of heads and tails. I'm talking about serial correlation. – wwl Oct 11 '17 at 13:29
  • Is there is no serial correlation if you get H,H,H,H,H,H,H,H,H,H,H,H ? – drhab Oct 11 '17 at 13:41

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