Question 4

We've shown that Kobe had some long shooting streaks, but are they long enough to support the belief that he had hot hands? What can we compare them to?

To answer these questions, let's return to the idea of independence. Two processes are independent if the outcome of one process doesn't effect the outcome of the second. If each shot that a player takes is an independent process, having made or missed your first shot will not affect the probability that you will make or miss your second shot.

A shooter with a hot hand will have shots that are not independent of one another. Specifically, if the shooter makes his first shot, the hot hand model says he will have a higher probability of making his second shot. Let's suppose for a moment that the hot hand model is valid for Kobe. During his career, the percentage of time Kobe makes a basket (i.e. his shooting percentage) is about 45%, or in probability notation: $$ P(\mathrm{shot}_1=H) = 0.45.$$

If Kobe has a hot hand (not independent shots), then the probability that he makes his second shot would go up given that he made the first shot: $$P(\mathrm{shot}_2=H~|~\mathrm{shot}_1=H) > 0.45.$$

Is this statement true?