The hot hand in three coin flips
Flip a fair coin three times. Look at every flip that comes immediately after a head and record what proportion of those flips are heads. (If no flip comes after a head, that is, the first two flips are both tails, discard the sequence.)
What is the average of this proportion over all sequences that are not discarded? Most people expect exactly one half. Compute it.
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There are only eight sequences. Write down the proportion for each.
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