document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=953766>Question 953766</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation:<BR>\n" );
document.write( "There are 45 multiple-choice questions, each of 4 choices. Only one of them is correct. Suppose a student picks all the choices randomly. What is the probability of him getting at least 40 questions correct?\r<BR>\n" );
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document.write( "Thank you!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #582503 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=582503\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the probability is very close to 0 that he will get at least 40 questions correct.\r<BR>\n" );
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document.write( "see the following picture:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/030703.jpg\" </>\r<BR>\n" );
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document.write( "the total probability is 1 as it should be.<BR>\n" );
document.write( "the probability from x = 40 to x = 45 is equal to 2.49889E-19.<BR>\n" );
document.write( "that's 2.49889 * 10^-19.<BR>\n" );
document.write( "it's so small that it is effectively 0.\r<BR>\n" );
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document.write( "you use the binomial theorem.\r<BR>\n" );
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document.write( "that theorem states:\r<BR>\n" );
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document.write( "p(x) = c(n,x) * p^x * q^(n-x)\r<BR>\n" );
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document.write( "to find the probability of getting more then or equal to 40 right, you need to take the sum of p(x) from x = 40 to x = 45.\r<BR>\n" );
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document.write( "that's what is shown in the picture.\r<BR>\n" );
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document.write( "one of those calculations would be done as follows:\r<BR>\n" );
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document.write( "p(40) = c(45,40) * .25^40 * .75^5.<BR>\n" );
document.write( "c(45,40) is equal to 1221759<BR>\n" );
document.write( ".25^40 is equal to 7,26179513... * 10^-25<BR>\n" );
document.write( ".75^5 is equal to .2373046875\r<BR>\n" );
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document.write( "c(45,40) is the number of possible ways you can get 40 items out of 45 if you don't take order into account.   That's the combination formula of C(n,x) = n! / (x! * (n-x)!)\r<BR>\n" );
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document.write( "c(45,40) * .25^40 * .75^5 is equal to 2.39823762 * 10^-19\r<BR>\n" );
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document.write( "the picture for x = 40 is showing you a probability of 2.39824E-19.\r<BR>\n" );
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document.write( "that's the probability of getting exactly 40 right out of the 45 questions/.\r<BR>\n" );
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document.write( "the picture value is rounded more than the calculated value.<BR>\n" );
document.write( "that's the only difference.<BR>\n" );
document.write( "they are the same.\r<BR>\n" );
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document.write( "the same formula is applied for x = {41, 42, 43, 44, 45} and then all of the probabilities from x = 40 to x = 45 are added up to get the probability of x being greater than or equal to 40.\r<BR>\n" );
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document.write( "it's a very small number that is, for all practical purposes, equal to 0.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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