Hot Streaks

Chance of having a 6-make streak out of 200 chances as a function of shooting percentage

I was playing around with streak probabilities, and it's remarkable how fast the probability of a certain streak rises with even a moderate increase in percentage per try.  Here, for example, is a toy model of a basketball player who takes 200 3-point shots in a given season.  Suppose we're interested in the chances that, at some point, the player has a "hot streak" of 6 makes in a row --- something that fans are bound to remember for some time.  If the player is a 30% shooter, there is only a 10% chance of such a streak occurring.  But the chances double for a 35% shooter, and for a very good shooter (45%) we'd expect such a streak more often than not.  I suspect these relatively small differences in skill are responsible for attributes of a so-called "streaky" shooter (which are almost never borne out by the statistics).  I chose a 6-shot streak arbitrarily, here is the likelihood for a 4-shot streak over the course of 50 attempts (a few games, so that this might happen a few times during the season, reinforcing the notion of "streaky"):

Chance of having a 4-make streak out of 50 chances as a function of shooting percentage

Here, you can see that for an average shooter (35%) you might expect a "hot streak" 40% of the time over a sample of 50 attempts.