SOLUTION: in a group of 25 students, 14 play tennis, 16 play gold, and 3 students play neither. what is the probability that a given student plays both tennis and golf?

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Question 544419: in a group of 25 students, 14 play tennis, 16 play gold, and 3 students play neither. what is the probability that a given student plays both tennis and golf?
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Two ways to do it:  (1) The formula way and 
(2) the Venn Diagram-Algebra way
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(1) The formula way:

P(T' and G') = 3%2F25

By DeMorgan's law,  T' and G' = (T or G)' , so

P[(T or G)'] = 3%2F25

P(T or G) = 1 - 3%2F25 = 25%2F25 - 3%2F25 = 22%2F25

P(T or G) = P(T) + P(G) - P(T and G)

22%2F25 = 14%2F25 + 16%2F25 - P(T and G)

P(T and G) = 14%2F25 + 16%2F25 - 22%2F25

P(T and G) = 8%2F25
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(2) the Venn Diagram-Algebra way
 



We have the system of equations:

x + y     = 14
    y + z = 16
x + y + z + 3 = 25

Solve that and get x = 6, y = 8, z = 8

So we have:



The one that play tennis and golf are in this part of the
Venn diagram, because it's part of both T and G:



That's 8 out of 25 which is 8%2F25

Edwin