A special basketball shot is made by James 20% of the time, by Kristy 30% of the time, and by Hannah 40% of the time. A game is played in which the players shoot consecutively and the first player to make the shot wins. If the order of shooting is first James, then Kristy, and then Hannah, which of the following statements are true?
- Kristy wins most often.
- Because Hannah is best at making this shot, she is more likely to win than
Kristy. - James will win least often.
- James, Kristy and Hannah all win the same number of games.
- James will win at least as often as either of the others.
- b, e
- d
- a, c
- b, c
- They are all false.
If you compute the probabilities of James, Kristy, or Hannah winning in the first round, second round, etc., it becomes clear that Kristy has the highest probability of winning in any given round, and James has the lowest. Thus, statements (a) and (c) are true and the answer is (3).
To illustrate, here are the probabilities for the first two rounds, with the last person listed being the winner:
J: James wins with probability .2 on the first shot.
JK: Kristy wins with probability .8(.3) = .24 on the second shot.
JKH: Hannah wins with probability .8(.7)(.4) = .224 on the third shot.
JKHJ: James wins with probability .8(.7)(.6)(.2) = .067 on the fourth shot.
JKHJK: Kristy wins with probability .8(.7)(.6)(.8)(.3) = .081 on the fifth shot.
JKHJKH: Hannah wins with probability .8(.7)(.6)(.8)(.7)(.4) = .075 on the sixth shot.
And so on . . .