Basketball is, of course, a game of both skill and chance. Is there some best way in which to throw the ball to increase the probability of making a basket? For example, is it better to toss the ball in a high arc or to throw it along a flatter trajectory? When might spin be beneficial, and when is it undesirable?

In a free throw (where a player gets an uncontested shot at the basket from about 4.3 meters), a player might employ the overhand push shot, in which the ball is pushed away from about shoulder height and then released. Instead, the player might use an underhand loop shot, in which the ball is brought upward from about the belt-line level and released. The first technique is the overwhelming choice among professional players, but the legendary Rick Barry set the record for free-throw shooting with the underhand technique. Does one technique actually provide a better chance at making a shot?

Answer From any position on the court, there is a wide range of angles at which you can launch the ball to send it through the basket, provided that you give the ball the proper speed. However, the fact that the ball is smaller in diameter than the basket allows a certain margin of error in the launch speed. If you choose a low angle, the margin for error is small and you must be quite accurate. You also must give the ball a large speed, which requires more force from you and which works against accuracy. If, instead, you choose an intermediate angle, the margin for error in the speed is larger, and the speed and force are smaller. So, you have a better chance at making the shot. For even larger angles, the margin for error is approximately the same, but the required speed and necessary force are larger, which makes larger angles less desirable.

Novice players usually shoot the ball along too flat a trajectory, but seasoned players learn through practice to arc the ball into the basket. The higher the shot is released, the slower the required launch must be, which gives an advantage to a tall player. The height advantage is so strong that some players elect to release the ball during a jump even when unchallenged by an opponent. If you put backspin on the ball and happen to hit the backboard instead of the basket, the spin creates friction that may cause the ball to rebound into the basket. When the shot is taken from one side, sidespin on the ball may also help.

The underhand free throw has a greater chance of success than the overhand throw, but the reasons are still debated. The success might be because the underhand throw is easier to execute, but a greater advantage seems to lie in the fact that the throw allows a player to put more backspin on the ball, which can make up for an errant toss onto the backboard.