The probability of a basketball player's making a free throw successfully at any time in a game is 1/2. if the player attempts ten free throws in a game, what is the probability that exactly six are made?

This is a problem in binomial probability.  Either the player is successful or he is not.  Here the number of "experiments" is 10, and the probability of "success" is 0.5.  

What is the probability that exactly six successes?

On my TI-83 Plus calculator, I've entered binompdf(10,0.5,6).  The result is 0.01898, or, rounded off, approx 0.019.  The chances that the player will score 6 times out of 10 is about 1.9%.