document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=485391>Question 485391</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Respected Sir / Mam ,<BR>\n" );
document.write( "Please help me to solve this question. I will be very grateful for your help .<BR>\n" );
document.write( "My question is <BR>\n" );
document.write( "If the overall percentage of success  in the exam is 60, what is the probability that out of a group of 4 students, at least one has passed?<BR>\n" );
document.write( "As the answer given  is : 0.9744<BR>\n" );
document.write( "Please provide me the steps for this question.\r<BR>\n" );
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document.write( "Thank you<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #331988 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=331988\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Like many things in this life there is a hard way and an easy way to answer this question.\r<BR>\n" );
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document.write( "If 4 students take the test, then there are 5 situations that could possibly arise:\r<BR>\n" );
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document.write( "None pass, 1 passes, 2 pass, 3 pass, or all 4 pass.\r<BR>\n" );
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document.write( "Note for future reference that since these five outcomes are inclusive of all possible outcomes, the sum of the probabilities of each must be 1, because it is a certainty that one of the outcomes will occur.\r<BR>\n" );
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document.write( "Now, your question asks for the probability that <i>at least one</i> student passes.  Looking at it in a straight-forward way, that would be the sum of the probabilities that <i>exactly</i> 1 passes plus <i>exactly</i> 2 pass plus...and so on.\r<BR>\n" );
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document.write( "Since \"Pass/Fail\" is an either/or outcome, and the probability of success for any given instance of one student taking one test is given as 60%, we know that the probability should be calculated using the binomial distribution.\r<BR>\n" );
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document.write( "The probability of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> successes in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> trials where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> is the probability of success on any given trial is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_n(k,p)\ =\ \left(n\cr k\right\)\left(p\right)^k\left(1\,-\,p\right)^{n\,-\,k}\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(n\cr k\right\)\"> is the number of combinations of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> things taken <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> at a time and is calculated by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{n!}{k!(n\,-\,k)!}\">\r<BR>\n" );
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document.write( "But you would need to perform this calculation four times and then sum the results.  First you would need to calculate:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_4(1,0.6)\ =\ \left(4\cr 1\right\)\left(0.6\right)^1\left(0.4\right)^{3}\">\r<BR>\n" );
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document.write( "To get the probability for exactly 1, and then you would have to do exactly 2, and so on.  In summary:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_4(\geq{1},0.6)\ =\ \sum_{i=1}^4\left(4\cr i\right\)\left(0.6\right)^i\left(0.4\right)^{n\,-\,i}\">\r<BR>\n" );
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document.write( "But fortunately, there is a much simpler way.  From the point of view of your question, there are only two possible outcomes.  Either 1 or more pass, or nobody passes.  So if you take the probability that nobody passes and subtract that from 1, you get the probability that at least 1 passes.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_4(1,0.6)\ =\ 1\ -\ P_4(0,0.6)\ =\ 1\ -\ \left(4\cr 0\right\)\left(0.6\right)^0\left(0.4\right)^{4}\">\r<BR>\n" );
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document.write( "Since we know that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> pick 0 is 1 for all positive integers <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> and that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a^0\ =\ 1\"> for all real numbers <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\">,\r<BR>\n" );
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document.write( "The above reduces to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 1\ -\ \left(0.4\right)^4\">\r<BR>\n" );
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document.write( "The arithmetic is yours to do.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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