document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206328>Question 1206328</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A coin is tossed 13 times.     <BR>\n" );
document.write( "a) How many different outcomes are possible?     <BR>\n" );
document.write( " b) What is the probability of getting exactly 3 heads?      <BR>\n" );
document.write( "c) What is the probability of getting at least 2 heads?     <BR>\n" );
document.write( " d) What is the probability of getting at most 9 heads?   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843695 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=843695\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> this looks like a binomial distribution type problem.\r<BR>\n" );
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document.write( "formula for that 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( "c(n,x) = n! / (x! * (n-x)!)\r<BR>\n" );
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document.write( "in this problem:<BR>\n" );
document.write( "n = 13<BR>\n" );
document.write( "x = 0 to 13<BR>\n" );
document.write( "p = probability of heads = .5<BR>\n" );
document.write( "q = probability of tails = .5\r<BR>\n" );
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document.write( "the total probability will be equal to 1.\r<BR>\n" );
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document.write( "the excel spreadsheet shown below gives you all the probabilities.\r<BR>\n" );
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document.write( "your solutions are:\r<BR>\n" );
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document.write( "sum of all probabilities equals 1.\r<BR>\n" );
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document.write( "probability of getting exactly 3 heads equals 0.034912109.\r<BR>\n" );
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document.write( "probability of getting at least 2 heads = 0.998291016.\r<BR>\n" );
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document.write( "probability of getting at most 9 heads = 0.953857422.\r<BR>\n" );
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document.write( "as an example of the calculations involved, we'll look at the probability of getting at least 2 heads.\r<BR>\n" );
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document.write( "since the probability of getting at least 2 heads is the same as 1 minus the probability of getting less than 2 heads, and it's much easier to make two calculations rather than 12 calculations, we'll calculate 1 minus the probability of getting less than 2 heads..\r<BR>\n" );
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document.write( "formula is p(x) = p^x * q^(n-x) * c(n,x).<BR>\n" );
document.write( "c(n,x) = n! / (x! * (n-x)!)\r<BR>\n" );
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document.write( "n = 13<BR>\n" );
document.write( "p = .5<BR>\n" );
document.write( "q = .5\r<BR>\n" );
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document.write( "p(x = 0) = .5^0 * .5^13 * c(13,0) = .0001220703125.<BR>\n" );
document.write( "p(x = 1) = .5^1 * .5^12 * c(13,1) = .0015869141.\r<BR>\n" );
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document.write( "p(x = 0 to 1) = .0017089844.<BR>\n" );
document.write( "1 minus that = .9982910156.<BR>\n" );
document.write( "round that to 9 decimal digits (same as excel), and you get .998291016.<BR>\n" );
document.write( "that's the same as excel provides you.\r<BR>\n" );
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document.write( "the solution made use of the fact that p(x >= 2) is the same as 1 minus p(x < 2).<BR>\n" );
document.write( "this allowed many less manual calculations to be performed.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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