Properties of Outcomes Practice Problems
Set 1: Properties of Outcomes—Nondisjoint Outcomes
Review the concept of nondisjoint outcomes at Properties of Outcomes Help
Practice
Imagine that a certain high school has 1000 students. The new swimming and diving coach, during his first day on the job, is looking for team prospects.
Suppose that the following are true:
 200 students can swim well enough to make the swimming team
 100 students can dive well enough to make the diving team
 30 students can make either team or both teams
If the coach wanders through the hallways blindfolded and picks a student at random, determine the probabilities, expressed as ratios, that the coach will pick
 a fast swimmer; call this p ( S )
 a good diver; call this p ( D )
 someone good at both swimming and diving; call this p ( S ∩ D )
 someone good at either swimming or diving, or both; call this p ( S ∪ D )
Solution
This problem is a little tricky. We assume that the coach has objective criteria for evaluating prospective candidates for his teams! That having been said, we must note that the outcomes are not mutually exclusive, nor are they independent. There is overlap, and there is interaction. We can find the first three answers immediately, because we are told the numbers:
p ( S ) = 200/1000 = 0.200
p ( D ) = 100/1000 = 0.100
p ( S ∩ D ) = 30/1000 = 0.030
In order to calculate the last answer – the total number of students who can make either team or both teams – we must find p ( S ∪ D ) using this formula:
p ( S ∪ D ) = p ( S ) + p ( D ) = p ( S ∩ D )
= 0.200 + 0.100 – 0.030 = 0.270
This means that 270 of the students in the school are potential candidates for either or both teams. The answer is not 300, as one might at first expect. That would be the case only if there were no students good enough to make both teams. We mustn’t count the exceptional students twice. (However well somebody can act like a porpoise, he or she is nevertheless only one person!)
Set 2: Properties of Outcomes—Multiple Outcomes
Review the concept of multiple outcomes at Properties of Outcomes Help.
Practice
Consider again the high school with 1000 students. The coach seeks people for the swimming, diving, and water polo teams in the same wandering, blindfolded way as before. Suppose the following is true of the students in the school:
 200 people can make the swimming team
 100 people can make the diving team
 150 people can make the water polo team
 30 people can make both the swimming and diving teams
 110 people can make both the swimming and water polo teams
 20 people can make both the diving and water polo teams
 10 people can make all three teams
If the coach staggers around and tags students at random, what is the probability, expressed as a ratio, that the coach will, on any one tag, select a student who is good enough for at least one of the sports?

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