document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169467>Question 1169467</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I need help solving this problem, I keep getting it incorrect.<BR>\n" );
document.write( "\"A student takes a 24-question multiple-choice exam but didn't study and randomly guesses each answer. Each question has three possible choices for the answer. Find the probability that the student guesses more than 75% of the questions correctly. (Round to three decimal places)\" </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #794221 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=794221\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> there are 24 questions.<BR>\n" );
document.write( "each question has 3 possible answers.<BR>\n" );
document.write( "you are looking for the probability that the student guesses more than 75% of the questions correctly.\r<BR>\n" );
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document.write( "the probability of getting any one question correct is equal to 1/3.\r<BR>\n" );
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document.write( "the probability of getting any one question wrong is equal to 2/3.\r<BR>\n" );
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document.write( "this looks like a binomial distribution type problem.\r<BR>\n" );
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document.write( "the formula is:\r<BR>\n" );
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document.write( "p(x) = p^x * q^(n-x) * c(n,x)\r<BR>\n" );
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document.write( "x is the probability of getting it right.<BR>\n" );
document.write( "q is the probability of getting it wrong.<BR>\n" );
document.write( "n is the size of the set of possible selections.<BR>\n" );
document.write( "x is the side of each subset of n.\r<BR>\n" );
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document.write( "when p = 1/3 and q = 2/3 and n = 24, then:\r<BR>\n" );
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document.write( "probability of getting more than 75% correct would be:\r<BR>\n" );
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document.write( "p(x) from x, where x > .75 * 24 = 3/4 * 24 = 18.\r<BR>\n" );
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document.write( "since 75% of 24 is 18, you would be looking for p(x) from x = 19 to 24.\r<BR>\n" );
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document.write( "the results are shown below:\r<BR>\n" );
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document.write( "the probability from x = 19 to x = 24 is shown as 5.4792E-06.<BR>\n" );
document.write( "that would be .0000054792.<BR>\n" );
document.write( "round that to 3 decimal places and it becomes 0.\r<BR>\n" );
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document.write( "an example of one of the calculations is:\r<BR>\n" );
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document.write( "the probability of getting exactly 19 correct is:<BR>\n" );
document.write( "p(19) = p^x * q^(n-1) * c(n,x) which becomes:<BR>\n" );
document.write( "p(19) = (1/3)^19 * (1/3)^5 * c(24,19) which becomes:<BR>\n" );
document.write( "p(19) = 4.81581359 * 10^6<BR>\n" );
document.write( "that compares favorably to 4.81581E-06 in the excel spreadsheet which is the same answer rounded to 5 decimal places.<BR>\n" );
document.write( "excel shows the rounded number, but uses the unrounded number.<BR>\n" );
document.write( "c(n,x) is the combination formula.<BR>\n" );
document.write( "for n = 24 and x = 19, the formula becomes c(24,19).<BR>\n" );
document.write( "c(n,x) formula is n! / (x! * (n-x)!)<BR>\n" );
document.write( "when n = 24 and x = 19, the formula becomes:<BR>\n" );
document.write( "24! / (19! * 5!)<BR>\n" );
document.write( "that can be written as (24 * 23 * 22 * 21 * 20 * 19!) / (19! * 5!)<BR>\n" );
document.write( "the 19! in the numerator and the denominator cancels out and you are left with:<BR>\n" );
document.write( "(24 * 23 * 22 * 21) / (5 * 4 * 3 * 2 * 1)<BR>\n" );
document.write( "solve to get c(14,19) = 42504<BR>\n" );
document.write( "that compares favorably to what is in the spreadsheet for p(19).\r<BR>\n" );
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document.write( "your results are the sum of p(19) to p(24).\r<BR>\n" );
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document.write( "the bottom line is, the probability of getting more than 75% correct is so low that it is effectively 0.\r<BR>\n" );
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document.write( "in excel, E-x means * 10^x.\r<BR>\n" );
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