document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1086784>Question 1086784</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me to solve this\r<BR>\n" );
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document.write( "a factory supervisor finds that 20 percent of the bolts produced by a machine will be defective . If 5 bolts are chosen at random,find the probability that at most 2 bolts will be defective.\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 #701020 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=701020\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Binomial Distribution:<BR>\n" );
document.write( "Probability of defect = p = 20% = 20/100 = 0.20 (use the decimal form)<BR>\n" );
document.write( "sample size = n = 5\r<BR>\n" );
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document.write( "We need to compute the probabilities for k = 0, k = 1, k = 2<BR>\n" );
document.write( "\"At most 2\" means \"2 is the ceiling\" or \"2 is the highest we can go\"\r<BR>\n" );
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document.write( "For k = 0, we would have this combination<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "5 C 0 = (5!)/(0!*(5-0)!)<BR>\n" );
document.write( "5 C 0 = (5!)/(0!*5!)<BR>\n" );
document.write( "5 C 0 = (1)/(0!*1)<BR>\n" );
document.write( "5 C 0 = (1)/(1)<BR>\n" );
document.write( "5 C 0 = 1\r<BR>\n" );
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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 = 0) = (5 C 0)*(0.2)^(0)*(1-0.2)^(5-0)<BR>\n" );
document.write( "P(X = 0) = (5 C 0)*(0.2)^(0)*(0.8)^(5)<BR>\n" );
document.write( "P(X = 0) = (1)*(0.2)^(0)*(0.8)^5<BR>\n" );
document.write( "P(X = 0) = (1)*(1)*(0.32768)<BR>\n" );
document.write( "P(X = 0) = 0.32768\r<BR>\n" );
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document.write( "For k = 1, we would have this combination<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "5 C 1 = (5!)/(1!*(5-1)!)<BR>\n" );
document.write( "5 C 1 = (5!)/(1!*4!)<BR>\n" );
document.write( "5 C 1 = (5*4!)/(1!*4!)<BR>\n" );
document.write( "5 C 1 = (5)/(1!)<BR>\n" );
document.write( "5 C 1 = (5)/(1)<BR>\n" );
document.write( "5 C 1 = 5/1<BR>\n" );
document.write( "5 C 1 = 5\r<BR>\n" );
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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 = 1) = (5 C 1)*(0.2)^(1)*(1-0.2)^(5-1)<BR>\n" );
document.write( "P(X = 1) = (5 C 1)*(0.2)^(1)*(0.8)^(4)<BR>\n" );
document.write( "P(X = 1) = (5)*(0.2)^(1)*(0.8)^4<BR>\n" );
document.write( "P(X = 1) = (5)*(0.2)*(0.4096)<BR>\n" );
document.write( "P(X = 1) = 0.4096\r<BR>\n" );
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document.write( "--------------------------------------------------------------------<BR>\n" );
document.write( "For k = 2, we would have this combination<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "5 C 2 = (5!)/(2!*(5-2)!)<BR>\n" );
document.write( "5 C 2 = (5!)/(2!*3!)<BR>\n" );
document.write( "5 C 2 = (5*4*3!)/(2!*3!)<BR>\n" );
document.write( "5 C 2 = (5*4)/(2!)<BR>\n" );
document.write( "5 C 2 = (5*4)/(2*1)<BR>\n" );
document.write( "5 C 2 = 20/2<BR>\n" );
document.write( "5 C 2 = 10\r<BR>\n" );
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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 = 2) = (5 C 2)*(0.2)^(2)*(1-0.2)^(5-2)<BR>\n" );
document.write( "P(X = 2) = (5 C 2)*(0.2)^(2)*(0.8)^(3)<BR>\n" );
document.write( "P(X = 2) = (10)*(0.2)^(2)*(0.8)^3<BR>\n" );
document.write( "P(X = 2) = (10)*(0.04)*(0.512)<BR>\n" );
document.write( "P(X = 2) = 0.2048\r<BR>\n" );
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document.write( "If we keep going, we would get this full Binomial distribution\r<BR>\n" );
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document.write( "<TABLE  border=1 cellpadding=3>\n" );
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document.write( "<tr><TD >X = k</TD><TD >P(X = k)</TD></tr>\r\n" );
document.write( "<tr><TD >0</TD><TD >0.32768</TD></tr>\r\n" );
document.write( "<tr><TD >1</TD><TD >0.4096</TD></tr>\r\n" );
document.write( "<tr><TD >2</TD><TD >0.2048</TD></tr>\r\n" );
document.write( "<tr><TD >3</TD><TD >0.0512</TD></tr>\r\n" );
document.write( "<tr><TD >4</TD><TD >0.0064</TD></tr>\r\n" );
document.write( "<tr><TD >5</TD><TD >0.00032</TD></tr>\r\n" );
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document.write( "</TABLE>\r<BR>\n" );
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document.write( "Now add up the probabilities for k = 0, k = 1, k = 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+%3C=+2%29+=+P%28X+=+0%29%2BP%28X+=+1%29%2BP%28X+=+2%29 ALIGN=MIDDLE  ALT=\"P%28X+%3C=+2%29+=+P%28X+=+0%29%2BP%28X+=+1%29%2BP%28X+=+2%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=+2%29+=+0.32768%2B0.4096%2B0.2048 ALIGN=MIDDLE  ALT=\"P%28X+%3C=+2%29+=+0.32768%2B0.4096%2B0.2048\"   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=+2%29+=+0.94208 ALIGN=MIDDLE  ALT=\"P%28X+%3C=+2%29+=+0.94208\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer: <font color=red>0.94208</font><BR>\n" );
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