Question 1151388
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 probability of S?
<pre>
Draw a Venn diagram with two sets, Q and S.  There are 4 regions
for the four possibilities:

Let w = P(Q occurs and not S does not occur)
Let x = P(Q occurs and S occurs)
Let y = P(S occurs and Q does not occur)
Let z = P(Q does not occur and S does not occur)

We put the 4 letters in the corresponding regions:

{{{drawing(300,200,-4,4,-2,4.8,
rectangle(-4,-1.6,4,4.4), locate(-2,1.8,w),locate(1.5,1.7,y),
locate(-3.7,-1,z),
locate(-3.6,2.5,Q), locate(-.1,1.8,x),
red(circle(-sqrt(2),sqrt(2),2)),
red(circle(-sqrt(2),sqrt(2),1.95)),
red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),
circle(sqrt(2),sqrt(2),1.95),
circle(sqrt(2),sqrt(2),1.975)),locate(3.4,2.5,S))}}}
</pre>the probability that Q occurs but S does not occur is 1/9.<pre> 
That is the part of the red circle Q that is not part of the blue
circle, the left part of the red circle Q. So w=1/9. So we put
1/9 in place of w.

{{{drawing(300,200,-4,4,-2,4.8,
rectangle(-4,-1.6,4,4.4), locate(-2,1.8,1/9),locate(1.5,1.7,y),
locate(-3.7,-1,z),
locate(-3.6,2.5,Q), locate(-.1,1.8,x),
red(circle(-sqrt(2),sqrt(2),2)),
red(circle(-sqrt(2),sqrt(2),1.95)),
red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975)),locate(3.4,2.5,S))}}}
</pre>the probability that at least one of them occurs is 1/3...<pre>
Therefore we have the equation:

{{{1/9+x+y=1/3}}}
</pre>What is the probability of S?<pre>
Since the probability of S is x+y, we can determine that by
solving for x+y:

{{{matrix(5,3,
1/9+x+y,""="",1/3,
x+y,""="",1/3-1/9,
x+y,""="",3/9-1/9,
x+y,""="",2/9,
"P(S)",""="",2/9 )}}}  <--answer

[We have the answer already. We did not need to use that they are independent.
In fact as it turns out, they could NOT be independent!!]

Edwin</pre>