From Drew Mathieson comes an exploration of basketball’s historied hot hand:

This season, on the way to winning her fourth WNBA championship in her 17-year career, Sue Bird made approximately 50 percent of her field goal attempts. Suppose she and Seattle Storm teammate Breanna Stewart are interested in testing whether Bird has a “hot hand” — that is, if her chances of making a basket depend on whether or not her previous shot went in. Bird happens to know that her chances of making any given shot is *always* 50 percent, independent of her shooting history, but she agrees to perform an experiment.

In each trial of the experiment, Bird will take three shots, while Stewart will record which shots Bird made or missed. Stewart will then look at all the trials that had at least one shot that was preceded by a made shot. She will randomly pick one of these trials and then randomly pick a shot that was preceded by a made shot. (If there was only one such shot to pick from, she will choose that shot.)

What is the probability that Bird made the shot that Stewart picked?