SOLUTION: Suppose Q and S are independent events such that the probability that at least one of them occurs is 1/3 and the probability that Q occurs but S does not occur is 1/9. What is the

Let P(S occurs) = x 
Let P(Q occurs) = y

Then 

P(S does not occur) = 1 - x
P(Q does not occur) = 1 - y
P(neither occurs) = (1 - x)(1 - y)


P(at least one occurs) = 1 - P(neither occurs) = 

P(at least one occurs) =   (given)

So  

P(Q occurs but S does not) = y(1-x)

P(Q occurs but S does not) =   (given)

So 

Therefore we have this system:



Simplifying the first:










Simplifying the second:





So the simplified system is:



Multiply the first equation through by -3:



Adding the equations term by term:





That's all that was asked for.  However
it you had been asked for the probability
of Q you would substitute  into














So  and 

Edwin