document.write( "Question 1048656: A survey of 100 students asked if they studied a foreign language. The result showed: Spanish, 28; German, 30; French, 42; Spanish and German, 8; Spanish and French,10; German and French 5; all three languages,3. What is the probability that a randomly selected student studied no foreign language? \n" ); document.write( "

Algebra.Com's Answer #664277 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "A survey of 100 students asked if they studied a foreign language. The result showed:
\n" ); document.write( "Spanish, 28; German, 30; French, 42; Spanish and German, 8; Spanish and French,10; German and French 5; all three languages,3.
\n" ); document.write( "What is the probability that a randomly selected student studied no foreign language?
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document.write( "Let S be the subset of the 100 students studying Spanish, and let |S| be the cardinality of the subset S. We are given |S| = 28.\r\n" );
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document.write( "Let G be the subset of the 100 students studying German, and let |G| be the cardinality of the subset G. We are given |G| = 30.\r\n" );
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document.write( "Let F be the subset of the 100 students studying French, and let |F| be the cardinality of the subset F. We are given |F| = 42.\r\n" );
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document.write( "Let SG be the subset of the 100 students studying Spanish and German. \r\n" );
document.write( "    It is the intersection of the sets S and G. \r\n" );
document.write( "    And let |SG| be the cardinality of the subset SG. We are given |SG| = 8.\r\n" );
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document.write( "Let SF be the subset of the 100 students studying Spanish and French. \r\n" );
document.write( "    It is the intersection of the sets S and F. \r\n" );
document.write( "    And let |SF| be the cardinality of the subset SF. We are given |SF| = 10.\r\n" );
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document.write( "Let GF be the subset of the 100 students studying German and French. \r\n" );
document.write( "    It is the intersection of the sets G and F. \r\n" );
document.write( "    And let |GF| be the cardinality of the subset GF. We are given |GF| = 5.\r\n" );
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document.write( "Finally, let SGF be the subset of the 100 students studying Spanish, German and French. \r\n" );
document.write( "    It is the intersection of the sets S, G and F. \r\n" );
document.write( "    And let |SGF| be the cardinality of the subset SGF. We are given |GF| = 3.\r\n" );
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document.write( "Now, the number of students among of 100 surveyed who studies at least one of these three languages is\r\n" );
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document.write( "   N = |S| + |G| +|F| - |SG| - |SF| - |GF| + |SGF| = 28 + 30 + 42 - 8 - 10 - 5 + 3 = 80.\r\n" );
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document.write( "Thus 80 students of 100 learn at least one language.\r\n" );
document.write( "The rest of 100, 100-80 = 20, do not study these languages.\r\n" );
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document.write( "Therefore, the probability that a randomly selected student studied no foreign language is  = .\r\n" );
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document.write( "Answer.  The probability that a randomly selected student studied no foreign language is .\r\n" );
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\n" ); document.write( "\n" ); document.write( "The formula we used is well known in the elementary set theory.\r
\n" ); document.write( "\n" ); document.write( "Its proof is very straightforward and simple.\r
\n" ); document.write( "\n" ); document.write( "See, for example, the lesson \r
\n" ); document.write( "\n" ); document.write( " - Advanced problems on counting elements in sub-sets of a given finite set\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( " - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lesson is the part of this online textbook in the topic \"Miscellaneous word problems\" of the section \"Word problems\".\r
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