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# Properties of Outcomes Help (page 3)

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By McGraw-Hill Professional
Updated on Aug 26, 2011

#### Practice 2

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?

#### Solution 2

Let the following expressions stand for the respective probabilities, all representing the results of random selections by the coach (and all of which we are told):

• Probability that a student can swim fast enough = p(S) = 200/ 1000 = 0.200.
• Probability that a student can dive well enough = p(D) = 100/ 1000 = 0.100.
• Probability that a student can play water polo well enough = p(W) = 150/1000 = 0.150.
• Probability that a student can swim fast enough and dive well enough = p(S D) = 30/1000 = 0.030.
• Probability that a student can swim fast enough and play water polo well enough = p(S W) = 110/1000 = 0.110.
• Probability that a student can dive well enough and play water polo well enough = p(D W) = 20/1000 = 0.020.
• Probability that a student can swim fast enough, dive well enough, and play water polo well enough = p(S D W) = 10/1000 = 0.010.

In order to calculate the total number of students who can end up playing at least one sport for this coach, we must find p(S D W) using this formula:

This means that 300 of the students in the school are potential prospects.

Practice problems for these concepts can be found at:

Basics of Probability Practice Test

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