Question 1124097
J, K, and L are events in sample space S. 
<pre>
Let's draw the Venn diagram with 3 sets where
the large rectangle represents the sample space S.
The letters "a" through "h" will represent the
probabilities of the regions in which the letters 
are located:

S
{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),   
circle(0,-.5,2),
locate(-2.3+.3,2,"a"),
locate(-3.5+.3,-2,"h"),
locate(0,-2.7,L),
locate(-.45+.3,-1,"g"),
locate(.6+.3,.4,"f"), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,J),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,K),
locate(-1.5+.3,.5,"d"),
locate(-.4+.3,2.3,"b"),
locate(1.8,2,"c"),
locate(-.4+.3,1.1,"e") )}}}

Pr(J) = a+b+d+e = 0.35 
Pr(K) = b+c+e+f = 0.18 
Pr(L) = d+e+f+g = 0.24 
Pr(J&#8745;K) = b+e = 0.11 
Pr(J&#8242;&#8745;L&#8242;) = c+h = 0.55 
Pr(K&#8242;&#8745;L) = d+g = 0.16

Also since S is the sample space:

Pr(S) = a+b+c+d+e+f+g+h = 1

We have this system of equations

a+b+c+d+e+f+g+h = 1.00
a+b  +d+e       = 0.35 
  b+c  +e+f     = 0.18 
      d+e+f+g   = 0.24 
  b    +e       = 0.11 
    c        +h = 0.55
      d    +g   = 0.16

By matrix methods using a TI-83 or TI-84,
we can solve for "a" through "g" in terms of "h"

a = 0.66-h
b = h-0.45
c = 0.55-h
d = h-0.42
e = 0.56-h
f = h-0.48
g = 0.58-h
</pre>What is Pr(J|K)?<pre>
Pr(J|K) = Pr(J&#8745;K)/Pr(K) = (b+e)/(b+c+e+f) =

[(h-0.45)+(0.56-h)]/[(h-0.45)+(0.55-h)+(0.56-h)+(h-0.48)] =

(h-0.45+0.56-h)/(h-0.45+0.55-h+0.56-h+h-0.48) = 

(-0.45+0.56)/(-0.45+0.55+0.56-0.48) =

0.11/0.18 = 11/18

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</pre>What is Pr(L|J)?<pre>
Pr(L|J) = Pr(L&#8745;J)/Pr(J) = (d+e)/(a+b+d+e) =

[(h-0.42)+(0.56-h)]/[(0.66-h)+(h-0.45)+(h-0.42)+(0.56-h)] =

(h-0.42+0.56-h)/(0.66-h+h-0.45+h-0.42+0.56-h) =

(-0.42+0.56)/(0.66-0.45-0.42+0.56) =

0.14/0.35 = 14/35 = 2/5

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</pre>What is Pr(K|L')?<pre>
Pr(K|L') = Pr(K&#8745;L')/Pr(L') = (b+c)/(a+b+c+h) =

[(h-0.45)+(0.55-h)]/[(0.66-h)+(h-0.45)+(0.55-h)+h] =

(h-0.45+0.55-h)/(0.66-h+h-0.45+0.55-h+h) = 

(-0.42+0.56)/(0.66-0.45-0.42+0.56) =

(-0.45+0.55)/(0.66-0.45+0.55)

0.10/0.76 = 10/76 = 5/38

Edwin</pre>