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We love the idea of the “phi bears” You could do this for any sorority that had a phi in it! I have found that dollar stores are perfect places to buy candy and little gifts to put into the baskets. A recent working paper has come out arguing for the existence of Hot Hands (in basketball), a concept psychologists had dismissed decades ago. That is, perhaps by selecting only heads values we are reducing the number of available heads. There is still a significant divergence from .5 even in the case of 60 coints being flipped!
We can see this by looking at our probability table with each sequence arranged by the first head up to that point.
Thus we can see that according to this, no matter that the sequence the likelihood of the next flip being a head after the first is .5.

To get some insight into what is happening, you have to think about how we are scoring each sequence. To leave a comment for the author, please follow the link and comment on their blog: Econometrics by Simulation. Big-Little Week has passed, but it is always good to keep these ideas in your back pocket for next year. Hot hands is where a player is thought to have a higher likelihood of scoring the next basket if the last three baskets where shot successfully. Knowing the previous outcome in the sequence does not provide information on future outcomes.
However, I don’t really know how well this sampling issue is known to time series people. In the case of HH we are asking what the next score is and rounding our previous positive result to match that.

I do not recall hearing of it in my meager studies of the matter, but frankly that could mean very little. This is because the selective scoring (last value scored) is only happening on a smaller and smaller portion of the series.
Thus an isolated few negative outcomes are kept while a few positive outcomes are downgraded resulting in an overall selection effect which reduces the likelihood of seeing a head overall after seeing a previous head.