document.write( "Question 1163288: In a high school class of 100 students, 42 studied mathematics, 68 studied
\n" ); document.write( "psychology, 54 studied history, 22 studied both mathematics and history, 25
\n" ); document.write( "studied both mathematics and psychology, 7 studied history and neither
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Algebra.Com's Answer #787343 by ikleyn(52781) $\"\"$ $\"About$

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In a high school class of 100 students, 42 studied mathematics, 68 studied
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\n" ); document.write( "studied both mathematics and psychology, 7 studied history and neither
\n" ); document.write( "mathematics nor psychology, 10 studied all three subjects and 8 did not take
\n" ); document.write( "any of the three. If a student is selected at random, find the probability that
\n" ); document.write( "(i) he takes history and psychology but not mathematics?
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document.write( "We have a universal set of 100 students and the following subsets\r\n" );
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document.write( "    M   studied Math         (42)\r\n" );
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document.write( "    P   studied Psychology   (68)\r\n" );
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document.write( "    H   studied History      (54)\r\n" );
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document.write( "    M ∩ H  studied Math and History     (22)\r\n" );
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document.write( "    M ∩ P  studied Math and Psychology  (25)\r\n" );
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document.write( "    Ho   studied History and neither Math nor Psychology  (7)   ( = H \ (H ∩ M) \ (H ∩ P) )\r\n" );
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document.write( "    M ∩ P ∩ H  studied all 3 subjects\r\n" );
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document.write( "    U \ (M U P U H)  did not take any of the three subjects  (8)\r\n" );
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document.write( "So, from the condition, the union  (M U P U H) of those who study at least one of the three subjects, has  100-8 = 92 students.\r\n" );
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document.write( "They want you determine the number of students who take History and Psychology but not Mathematics.\r\n" );
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document.write( "This set is  (H ∩ P) \ (H ∩ P ∩ M).\r\n" );
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document.write( "To solve the problem, we will find first the number of students in the set  (H ∩ P), and then will subtract 10 (which is  (H ∩ P ∩ M) )  from it.\r\n" );
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document.write( "To find the number of students in the set  (H ∩ P), use this remarkable identity from the elementary set theory\r\n" );
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document.write( "    n(M U P U H) = n(M) + n(P) + n(H) - n(M ∩ P) - n(M ∩ H) - n(P ∩ H) + n(M ∩ P ∩ H).    (1)\r\n" );
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document.write( "This identity says that the number of element in the union of any three subsets of a universal set is the sum of elements \r\n" );
document.write( "in each subset minus the sum of elements in in-pair intersections of the subsets plus the number of elements in the triple \r\n" );
document.write( "intersection of the three subsets.\r\n" );
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document.write( "This identity is valid for any three subsets of a universal set.\r\n" );
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document.write( "Substitute the given value into the identity (1).  You will get then\r\n" );
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document.write( "    92 = 42 + 68 + 54 - 22 - 25 - n(H ∩ P) + 10.\r\n" );
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document.write( "It implies \r\n" );
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document.write( "    n(H ∩ P) = 42 + 68 + 54 - 22 - 25 + 10 - 92 = 35.\r\n" );
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document.write( "The last step to get the answer is to subtract 10 from 35:  \r\n" );
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document.write( "    the number of students who take History and Psychology but not Mathematics = n((H ∩ P) - n(M ∩ P ∩ H) = 35 - 10 = 25.    \r\n" );
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document.write( "So the probability the problem asks for is  P =  = 0.25.    ANSWER\r\n" );
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