Question 1124097
.
<pre>
J, K, and L are events in sample space S. 
Pr(J)=0.35 
Pr(K)=0.18 
Pr(L)=0.24 
Pr(J&#8745;K)=0.11 
Pr(J&#8242;&#8745;L&#8242;)=0.55      (*)
Pr(K&#8242;&#8745;L)=0.16       (**)

What is Pr(J|K)? 

What is Pr(L|J)? 
 
What is Pr(K|L&#8242;)?</pre>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Although the answer produced by Edwin is correct, I do not think 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;that his approach and his solution are adequate and that they are how they expected to be.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I put here my solution, &nbsp;and after completing it, &nbsp;I put my reasons why I think the Edwin' &nbsp;solution and approach are inadequate. 




1. &nbsp;&nbsp;<U>What is Pr(J|K) ?</U>


<pre>
    Pr(J|K) = by the definition = Pr(J&#8745;K)/Pr(K) = {{{0.11/0.18}}} = {{{11/18}}}.
</pre>


2. &nbsp;&nbsp;<U>What is Pr(L|J) ?</U>


<pre>
    In the "given" part, find a line (*) related to L and J.

    This line is  Pr(J' &#8745; L')=0.55

    From the elementary set theory,  J' &#8745; L' = (S\J) &#8745; (S\L) = S \ (J U L).   (Symbol " \" means subtraction of sets.)

        You can prove it on your own using (or without using) Venn diagram for 2 subsets J and L in S.


    Also, from the basics of probability,  Pr(J U L) = Pr(J) + Pr(L) - Pr((J &#8745; L).


                These pieces are <U>THE KEY part of the solution</U>, which you <U>SHOULD and MUST understand</U> <U>when solving such problems</U> !


    Therefore,  0.55 = Pr(J' &#8745; L') = 1 - (Pr(J) + Pr(L)) + Pr((J &#8745; L).  

    which implies  Pr(J &#8745; L) = 0.55 + Pr(J) + Pr(L) - 1 = 0.55 + 0.35 + 0.24 - 1 = 0.14.


    Therefore, 

    Pr(L|J) = by the definition = Pr(L&#8745;J)/Pr(K) = {{{0.14/0.35}}} = {{{2/5}}}.
</pre>


3. &nbsp;&nbsp;<U>What is Pr(K|L')?</U>


<pre>
    By the definition,  Pr(K|L') = P(K &#8745; L')/P(L'). 

    Pr(L') = 1 - Pr(L) = 1 - 0.24 = 0.76.

    What is  P(K &#8745; L') ?


    In the "given" part, find a line (**) related to K and L.

    This line is  Pr(K' &#8745; L)=0.16.


     The set   (K' &#8745; L)  is  "L without K",  or,  in other words,  L\ (L &#8745; K).

         Hence,  Pr((K' &#8745; L) = 0.16 = Pr(L) - Pr(L &#8745; K),   which gives

             Pr(L &#8745; K) = Pr(L) - 0.16 = 0.024 - 0.16 = 0.08. 


     The set   (K &#8745; L')  is  "K without L",  or,  in other words,  K\ (L &#8745; K).

         Hence,  Pr(K &#8745; L') = 0.18 - Pr(L &#8745; K) = 0.18 - 0.08 = 0.10.


     Thus finally  Pr(K|L') = {{{0.10/0.76}}} = {{{5/38}}}.
</pre>

So my answers coincide with that by Edwin.   &nbsp;&nbsp;Which is good.


This problem is not of the school Math level.  &nbsp;It is of the level of introductory problems to a Probability theory course 
at an advanced Math college or a renowned University.


And the goal of such exercises is to teach the students to basic operations on the sets and to connect it with the basics of Probability.


It is an approach I followed in my solution.


Every of the three questions of this assignment should be answered based o information related to relevant two sets of each question.


There is no any need to involve the third set.


There is no any need to use Venn diagram for three sets.


There is no any need to construct the system of 7 equations and solve it.


It is why I chose another, &nbsp;more adequate approach and wrote this post.


---------------------


I am not seeking for a war with Edwin.


In opposite, &nbsp;I esteem him as a great tutor at this forum,  &nbsp;and, &nbsp;probably, &nbsp;had I not see his post, &nbsp;I would not write this one.