Many basketball players and fans believe in the "hot hand." That is, they believe that a player is more likely to make a shot if that player has made several shots in a row
What does the hot hand hypothesis have to do with the idea of independent events? How might you test the hot hand hypothesis?
If you believe in the hot hand, then you believe that shooting a basketball several times are not independent events. And so you might, for example, believe that the probability of making a shot is higher if the player has made several shots in a row (the player is hot) than if the player had missed the last several shots. This is a testable hypothesis. You could gather data on shooting from games that have been played or by running an experiment to see if the probability a player makes a shot is independent of whether or not she made her last shot. The available evidence does not lend much support to the hot hand hypothesis.
See Thomas Gilovich, Robert Vallone, and Amos Tversky, "The Hot Hand in Basketball: On the Misperception of Random Sequences, Cognitive Psychology, 1985 (http://psych.cornell.edu/sites/default/files/Gilo.Vallone.Tversky.pdf)
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