Even though you are not to use a Venn diagram, use one privately anyway to get
the idea right and then set up the equations. Don't turn in the Venn diagram.



Let a = n(SᑎH')
Let b = n(SᑎH)
Let c = n(S'ᑎH)
Let d = n(S'ᑎH')

>>A survey of 448 students at a small college...<<

a+b+c+d = 448 = n(U) = n(SᑎH') + n(SᑎH) + n(S'ᑎH) + n(S'ᑎH')

>>...showed that 234 students responded that they did not like to ski,...<<

c+d = 234 = n(S'ᑎH) + n(S'ᑎH')

>>100 students responded that they played hockey,<< 

b+c = 100 =n(SᑎH) + n(S'ᑎH) 

>>and that there were 189 students the neither skied nor played hockey.<< 

d = 189 = n(S'ᑎH')

So we have the system of equations:



Solve that system by substitution (that's easy), getting 
a=159, b=55, c=45, d=189



Then the desired probability is 

b/(a+b+c+d) = n(SᑎH)/n(U) = 55/448

Edwin