document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1086877>Question 1086877</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 65% of men consider themselves football fans,you randomly select 8 men and ask each if he is a football fan.Find the probability that<BR>\n" );
document.write( "1.at least 5 are football fans<BR>\n" );
document.write( "2.less than five are football fans.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #701119 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=701119\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Let's compute the binomial probabilities for k = 5 through k = 8 (where k is an integer)<BR>\n" );
document.write( "For each case, the probability of success is p = 0.65 and the sample size is n = 8\r<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------------------------<BR>\n" );
document.write( "For k = 5, we have this combination<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "8 C 5 = (8!)/(5!*(8-5)!)<BR>\n" );
document.write( "8 C 5 = (8!)/(5!*3!)<BR>\n" );
document.write( "8 C 5 = (8*7*6*5!)/(5!*3!)<BR>\n" );
document.write( "8 C 5 = (8*7*6)/(3!)<BR>\n" );
document.write( "8 C 5 = (8*7*6)/(3*2*1)<BR>\n" );
document.write( "8 C 5 = 336/6<BR>\n" );
document.write( "8 C 5 = 56\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Leading to this binomial probability<BR>\n" );
document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)<BR>\n" );
document.write( "P(X = 5) = (8 C 5)*(0.65)^(5)*(1-0.65)^(8-5)<BR>\n" );
document.write( "P(X = 5) = (8 C 5)*(0.65)^(5)*(0.35)^(3)<BR>\n" );
document.write( "P(X = 5) = (56)*(0.65)^(5)*(0.35)^3<BR>\n" );
document.write( "P(X = 5) = (56)*(0.1160290625)*(0.042875)<BR>\n" );
document.write( "P(X = 5) = 0.2785857790625\r<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------------------------<BR>\n" );
document.write( "For k = 6, we have this combination<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "8 C 6 = (8!)/(6!*(8-6)!)<BR>\n" );
document.write( "8 C 6 = (8!)/(6!*2!)<BR>\n" );
document.write( "8 C 6 = (8*7*6!)/(6!*2!)<BR>\n" );
document.write( "8 C 6 = (8*7)/(2!)<BR>\n" );
document.write( "8 C 6 = (8*7)/(2*1)<BR>\n" );
document.write( "8 C 6 = 56/2<BR>\n" );
document.write( "8 C 6 = 28\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Leading to this binomial probability<BR>\n" );
document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)<BR>\n" );
document.write( "P(X = 6) = (8 C 6)*(0.65)^(6)*(1-0.65)^(8-6)<BR>\n" );
document.write( "P(X = 6) = (8 C 6)*(0.65)^(6)*(0.35)^(2)<BR>\n" );
document.write( "P(X = 6) = (28)*(0.65)^(6)*(0.35)^2<BR>\n" );
document.write( "P(X = 6) = (28)*(0.075418890625)*(0.1225)<BR>\n" );
document.write( "P(X = 6) = 0.25868679484375\r<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------------------------<BR>\n" );
document.write( "For k = 7, we have this combination<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "8 C 7 = (8!)/(7!*(8-7)!)<BR>\n" );
document.write( "8 C 7 = (8!)/(7!*1!)<BR>\n" );
document.write( "8 C 7 = (8*7!)/(7!*1!)<BR>\n" );
document.write( "8 C 7 = (8)/(1!)<BR>\n" );
document.write( "8 C 7 = (8)/(1)<BR>\n" );
document.write( "8 C 7 = 8/1<BR>\n" );
document.write( "8 C 7 = 8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Leading to this binomial probability<BR>\n" );
document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)<BR>\n" );
document.write( "P(X = 7) = (8 C 7)*(0.65)^(7)*(1-0.65)^(8-7)<BR>\n" );
document.write( "P(X = 7) = (8 C 7)*(0.65)^(7)*(0.35)^(1)<BR>\n" );
document.write( "P(X = 7) = (8)*(0.65)^(7)*(0.35)^1<BR>\n" );
document.write( "P(X = 7) = (8)*(0.04902227890625)*(0.35)<BR>\n" );
document.write( "P(X = 7) = 0.1372623809375\r<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------------------------<BR>\n" );
document.write( "For k = 8, we have this combination<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "8 C 8 = (8!)/(8!*(8-8)!)<BR>\n" );
document.write( "8 C 8 = (8!)/(8!*0!)<BR>\n" );
document.write( "8 C 8 = (1)/(1*0!)<BR>\n" );
document.write( "8 C 8 = (1)/(1*1)<BR>\n" );
document.write( "8 C 8 = (1)/(1)<BR>\n" );
document.write( "8 C 8 = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Leading to this binomial probability<BR>\n" );
document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)<BR>\n" );
document.write( "P(X = 8) = (8 C 8)*(0.65)^(8)*(1-0.65)^(8-8)<BR>\n" );
document.write( "P(X = 8) = (8 C 8)*(0.65)^(8)*(0.35)^(0)<BR>\n" );
document.write( "P(X = 8) = (1)*(0.65)^(8)*(0.35)^0<BR>\n" );
document.write( "P(X = 8) = (1)*(0.0318644812890625)*(1)<BR>\n" );
document.write( "P(X = 8) = 0.0318644812890625\r<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "We do not need to compute the probabilities for k = 0 to k = 4, but for the sake of completeness the probabilities are shown in the table below\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Full Binomial Distribution Table<BR>\n" );
document.write( "<TABLE  border=1 cellpadding=3>\n" );
document.write( "\r\n" );
document.write( "<tr><TD >X = k</TD><TD >P(X = k)</TD></tr>\r\n" );
document.write( "<tr><TD >0</TD><TD >0.0002251875390625</TD></tr>\r\n" );
document.write( "<tr><TD >1</TD><TD >0.0033456434375</TD></tr>\r\n" );
document.write( "<tr><TD >2</TD><TD >0.02174668234375</TD></tr>\r\n" );
document.write( "<tr><TD >3</TD><TD >0.0807733915625</TD></tr>\r\n" );
document.write( "<tr><TD >4</TD><TD >0.187509658984375</TD></tr>\r\n" );
document.write( "<tr><TD >5</TD><TD >0.2785857790625</TD></tr>\r\n" );
document.write( "<tr><TD >6</TD><TD >0.25868679484375</TD></tr>\r\n" );
document.write( "<tr><TD >7</TD><TD >0.1372623809375</TD></tr>\r\n" );
document.write( "<tr><TD >8</TD><TD >0.0318644812890625</TD></tr>\r\n" );
document.write( "\n" );
document.write( "</TABLE><BR>\n" );
document.write( "--------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We'll use that table to answer problem 1. Add up the probabilities for k = 5, k = 6, k = 7, k = 8 to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+%3E=+5%29+=+P%28X+=+5%29%2BP%28X+=+6%29%2BP%28X+=+7%29%2BP%28X+=+8%29 ALIGN=MIDDLE  ALT=\"P%28X+%3E=+5%29+=+P%28X+=+5%29%2BP%28X+=+6%29%2BP%28X+=+7%29%2BP%28X+=+8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+%3E=+5%29+=+0.2785857790625%2B0.25868679484375%2B0.1372623809375%2B0.0318644812890625 ALIGN=MIDDLE  ALT=\"P%28X+%3E=+5%29+=+0.2785857790625%2B0.25868679484375%2B0.1372623809375%2B0.0318644812890625\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+%3E=+5%29+=+0.70639943613281 ALIGN=MIDDLE  ALT=\"P%28X+%3E=+5%29+=+0.70639943613281\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Answer for problem one: <font color=red>0.70639943613281</font><BR>\n" );
document.write( "Round that however you need to\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The events \"less than 5\" and \"at least 5\" are complementary. One or the other must happen. Therefore their probabilities add to 1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+%3C+5%29+%2B+P%28X+%3E=+5%29+=+1 ALIGN=MIDDLE  ALT=\"P%28X+%3C+5%29+%2B+P%28X+%3E=+5%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+%3C+5%29+=+1-P%28X+%3E=+5%29 ALIGN=MIDDLE  ALT=\"P%28X+%3C+5%29+=+1-P%28X+%3E=+5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use the result from problem 1 to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+%3C+5%29+=+1-P%28X+%3E=+5%29 ALIGN=MIDDLE  ALT=\"P%28X+%3C+5%29+=+1-P%28X+%3E=+5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+%3C+5%29+=+1-0.70639943613281 ALIGN=MIDDLE  ALT=\"P%28X+%3C+5%29+=+1-0.70639943613281\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+%3C+5%29+=+0.29360056386719 ALIGN=MIDDLE  ALT=\"P%28X+%3C+5%29+=+0.29360056386719\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Answer for problem two: <font color=red>0.29360056386719</font><BR>\n" );
document.write( "Round that however you need to</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );