document.write( "Question 1124097: J, K, and L are events in sample space S.
\n" ); document.write( "Pr(J)=0.35
\n" ); document.write( "Pr(K)=0.18
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\n" ); document.write( "Pr(J∩K)=0.11
\n" ); document.write( "Pr(J′∩L′)=0.55
\n" ); document.write( "Pr(K′∩L)=0.16 \r
\n" ); document.write( "\n" ); document.write( "What is Pr(J|K)?
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\n" ); document.write( "\n" ); document.write( "What is Pr(L|J)?
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\n" ); document.write( "\n" ); document.write( "What is Pr(K|L′)?
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\n" ); document.write( "\n" ); document.write( "Enter your answers as whole numbers or fractions in lowest terms. \n" ); document.write( "
Algebra.Com's Answer #740659 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "J, K, and L are events in sample space S. \r\n" );
document.write( "Pr(J)=0.35 \r\n" );
document.write( "Pr(K)=0.18 \r\n" );
document.write( "Pr(L)=0.24 \r\n" );
document.write( "Pr(J∩K)=0.11 \r\n" );
document.write( "Pr(J′∩L′)=0.55      (*)\r\n" );
document.write( "Pr(K′∩L)=0.16       (**)\r\n" );
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document.write( "What is Pr(J|K)? \r\n" );
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document.write( "What is Pr(L|J)? \r\n" );
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document.write( "What is Pr(K|L′)?
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\n" ); document.write( "\n" ); document.write( "            Although the answer produced by Edwin is correct, I do not think
\n" ); document.write( "            that his approach and his solution are adequate and that they are how they expected to be.\r
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\n" ); document.write( "\n" ); document.write( "            I put here my solution,  and after completing it,  I put my reasons why I think the Edwin'  solution and approach are inadequate. \r
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\n" ); document.write( "\n" ); document.write( "1.   What is Pr(J|K) ?\r
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document.write( "    Pr(J|K) = by the definition = Pr(J∩K)/Pr(K) = \"0.11%2F0.18\" = \"11%2F18\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "2.   What is Pr(L|J) ?\r
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document.write( "    In the \"given\" part, find a line (*) related to L and J.\r\n" );
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document.write( "    This line is  Pr(J' ∩ L')=0.55\r\n" );
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document.write( "    From the elementary set theory,  J' ∩ L' = (S\J) ∩ (S\L) = S \ (J U L).   (Symbol \" \\" means subtraction of sets.)\r\n" );
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document.write( "        You can prove it on your own using (or without using) Venn diagram for 2 subsets J and L in S.\r\n" );
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document.write( "    Also, from the basics of probability,  Pr(J U L) = Pr(J) + Pr(L) - Pr((J ∩ L).\r\n" );
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document.write( "                These pieces are THE KEY part of the solution, which you SHOULD and MUST understand when solving such problems !\r\n" );
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document.write( "    Therefore,  0.55 = Pr(J' ∩ L') = 1 - (Pr(J) + Pr(L)) + Pr((J ∩ L).  \r\n" );
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document.write( "    which implies  Pr(J ∩ L) = 0.55 + Pr(J) + Pr(L) - 1 = 0.55 + 0.35 + 0.24 - 1 = 0.14.\r\n" );
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document.write( "    Therefore, \r\n" );
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document.write( "    Pr(L|J) = by the definition = Pr(L∩J)/Pr(K) = \"0.14%2F0.35\" = \"2%2F5\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "3.   What is Pr(K|L')?\r
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document.write( "    By the definition,  Pr(K|L') = P(K ∩ L')/P(L'). \r\n" );
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document.write( "    Pr(L') = 1 - Pr(L) = 1 - 0.24 = 0.76.\r\n" );
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document.write( "    What is  P(K ∩ L') ?\r\n" );
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document.write( "    In the \"given\" part, find a line (**) related to K and L.\r\n" );
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document.write( "    This line is  Pr(K' ∩ L)=0.16.\r\n" );
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document.write( "     The set   (K' ∩ L)  is  \"L without K\",  or,  in other words,  L\ (L ∩ K).\r\n" );
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document.write( "         Hence,  Pr((K' ∩ L) = 0.16 = Pr(L) - Pr(L ∩ K),   which gives\r\n" );
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document.write( "             Pr(L ∩ K) = Pr(L) - 0.16 = 0.024 - 0.16 = 0.08. \r\n" );
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document.write( "     The set   (K ∩ L')  is  \"K without L\",  or,  in other words,  K\ (L ∩ K).\r\n" );
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document.write( "         Hence,  Pr(K ∩ L') = 0.18 - Pr(L ∩ K) = 0.18 - 0.08 = 0.10.\r\n" );
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document.write( "     Thus finally  Pr(K|L') = \"0.10%2F0.76\" = \"5%2F38\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "So my answers coincide with that by Edwin.   Which is good.\r
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\n" ); document.write( "\n" ); document.write( "This problem is not of the school Math level.  It is of the level of introductory problems to a Probability theory course
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\n" ); document.write( "\n" ); document.write( "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.\r
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\n" ); document.write( "\n" ); document.write( "It is an approach I followed in my solution.\r
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\n" ); document.write( "\n" ); document.write( "Every of the three questions of this assignment should be answered based o information related to relevant two sets of each question.\r
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\n" ); document.write( "\n" ); document.write( "There is no any need to involve the third set.\r
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\n" ); document.write( "\n" ); document.write( "There is no any need to use Venn diagram for three sets.\r
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\n" ); document.write( "\n" ); document.write( "There is no any need to construct the system of 7 equations and solve it.\r
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\n" ); document.write( "\n" ); document.write( "It is why I chose another,  more adequate approach and wrote this post.\r
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\n" ); document.write( "\n" ); document.write( "I am not seeking for a war with Edwin.\r
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\n" ); document.write( "\n" ); document.write( "In opposite,  I esteem him as a great tutor at this forum,  and,  probably,  had I not see his post,  I would not write this one.\r
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