document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=296701>Question 296701</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A) A test consists of 10 true and false questions. To pass the test a student must answer at least  eight questions correctly. If the student guesses on each questions what is the probability that the student will pass the test? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #213773 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=213773\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Assume it was only 3 answers on the test and the student needed to get at least 2 out of 3 correct.\r<BR>\n" );
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document.write( "Probability of getting exactly 0 wrong would be .5^3 = .125 * 1 = .125<BR>\n" );
document.write( "Probability of getting exactly 1 wrong would be .5^3 = .125 * 3 = .375<BR>\n" );
document.write( "Probability of getting exactly 2 wrong would be .5^3 = .125 * 3 = .375<BR>\n" );
document.write( "Probability of getting exactly 3 wrong would be .5^3 = .125 * 1 = .125\r<BR>\n" );
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document.write( "Total probability is equal to 1 as it should be.\r<BR>\n" );
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document.write( "Probability of getting 0 or 1 wrong would be .375 + .125 = .5\r<BR>\n" );
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document.write( "Since 0 or 1 wrong is the same as getting 2 or 3 right, then this is the probability that the student will get at least 2 right.\r<BR>\n" );
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document.write( "The individual probabilities are multiplied by the number of ways they can occur.\r<BR>\n" );
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document.write( "If we let 0 = wrong and 1 = correct, then:\r<BR>\n" );
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document.write( "You can get 0 wrong only 1 way (111)<BR>\n" );
document.write( "You can get 1 wrong 3 ways (110) (101) (011)<BR>\n" );
document.write( "You can get 2 wrong 3 ways (001) (010) (100)<BR>\n" );
document.write( "You can get 3 wrong 1 way (000)\r<BR>\n" );
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document.write( "The same concept applies to the larger numbers.\r<BR>\n" );
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document.write( "With 10 answers, this is what happens:\r<BR>\n" );
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document.write( "p(0) = probability of getting exactly 0 correct.<BR>\n" );
document.write( "p(1) = probability of getting exactly 1 correct.<BR>\n" );
document.write( "etc.\r<BR>\n" );
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document.write( "p(0) = .5^10 = .000976563 * 1 = .000976563<BR>\n" );
document.write( "p(1) = .5^10 = .000976563 * 10 = .009765625<BR>\n" );
document.write( "p(2) = .5^10 = .000976563 * 45 = .043945313<BR>\n" );
document.write( "p(3) = .5^10 = .000976563 * 120 = .1171875<BR>\n" );
document.write( "p(4) = .5^10 = .000976563 * 210 = .205078125<BR>\n" );
document.write( "p(5) = .5^10 = .000976563 * 252 = .24609375<BR>\n" );
document.write( "p(6) = .5^10 = .000976563 * 210 = .205078125<BR>\n" );
document.write( "p(7) = .5^10 = .000976563 * 120 = .1171875<BR>\n" );
document.write( "p(8) = .5^10 = .000976563 * 45 = .043945313<BR>\n" );
document.write( "p(9) = .5^10 = .000976563 * 10 = .009765625<BR>\n" );
document.write( "p(10) = .5^10 = .000976563 * 1 = .000976563\r<BR>\n" );
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document.write( "Total probability equals 1 as it should.\r<BR>\n" );
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document.write( "Probability of getting 0 or 1 or 2 wrong is equal to:\r<BR>\n" );
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document.write( ".000976563 + .009765625 + .043945313 = .0546875\r<BR>\n" );
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document.write( "Since the probability of getting 0 or 1 or 2 wrong is the same as the probability of getting 8 or 9 or 10 right, then the probability that the student will get at least 8 correct is equal to .0546875.\r<BR>\n" );
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document.write( "The number of ways each percentage can be achieved is given by the formula:\r<BR>\n" );
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document.write( "For example, with 10 answers, the number of ways of getting exactly 4 wrong is equal to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810%21%29%2F%28%284%21%29%2A%286%21%29%29+=+%2810%2A9%2A8%2A7%2A6%21%29%2F%284%2A3%2A2%2A1%2A6%21%29+=+%285040%2A6%21%29%2F%2824%2A6%21%29+=+5050%2F24+=+210 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "With 10 answers, the number of ways of getting exactly 6 wrong is the same as getting exactly 4 wrong as shown below:\r<BR>\n" );
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document.write( " </SPAN>\n" );
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