Former professional basketball player Shaquille O'Neal was a notoriously poor free-throw shooter, with a career success rate of 52.7%. This was such a low percentage that opposing teams would often deliberately foul O'Neal to force him to shoot free throws, rather than risk his team scoring a more traditional basket.
O'Neal averaged 9.7 free-throw attempts per game.
Suppose each free-throw attempt is independent, and the probability of making a free throw is 0.527.
1. Let X be a random variable representing the number of successful free throws O'Neal makes. Find μχ, the expected number of successful free throws O'Neal makes in a game. Round to the nearest hundredth, if necessary.
2. The most free throws O'Neal ever attempted in a single game was 31.
(a) O'Neal made exactly 19 free throws in this game. Find the probability of O'Neal making exactly 19 of 31 free throws.
(b) Find the probability that O'Neal misses all 31 free throws.
(c) Find the probability that O'Neal makes all 31 free throws.
(d) Find the probability that O'Neal makes a majority of his attempts in this game.
(e) What is the expected number of free throws for O'Neal in this game (on exactly 31 attempts)? Round to the nearest tenth, if necessary.
3. Suppose Shaquille O'Neal has brought his team (the Lakers) to the NBA playoffs. His team is playing a best-of-seven series, where the first team to win four games wins the series.
Historically, O'Neal's teams had a playoff winning percentage of 59.7%. Assume, then, that 0.597 is the independent probability of the Lakers winning any individual playoff game.
(a) What is the probability that the Lakers win the series (are the first team to eventually win four games)?
(b) What is the probability that all seven games in the series need to be played in order to determine the winner? Note that this question is not concerned with the overall winner of the series, only that the series requires all seven games.
