document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1125355>Question 1125355</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would I compute this problem by hand; what is the formula?\r<BR>\n" );
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document.write( "A stationary store has decided to accept a large shipment of ball-point pens if an inspection of 17 randomly selected pens yields no more than two defective pens.\r<BR>\n" );
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document.write( "(a) Find the probability that this shipment is accepted if 5% of the total shipment is defective. (Use 3 decimal<BR>\n" );
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document.write( "(b) Find the probability that this shipment is not accepted if 15% of the total shipment is defective. (Use 3 decimal<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 #741858 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=741858\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> this looks like a binomial probability type of problem.\r<BR>\n" );
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document.write( "the sample size is 17.\r<BR>\n" );
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document.write( "the shipment is accepted if the number of defective pens is less than or equal to 2.\r<BR>\n" );
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document.write( "otherwise, it is not accepted.\r<BR>\n" );
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document.write( "if 5% of the shipment is defective, then the probability of randomly picking 1 defective pen out of the shipment is .05.\r<BR>\n" );
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document.write( "the binomial probability formula is p(x) = p^x * q^(n-x) * c(n,x)\r<BR>\n" );
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document.write( "p(x) is the probability of getting x defective.<BR>\n" );
document.write( "p is the probability that a randomly selected pen will be defective.<BR>\n" );
document.write( "q is equal to 1 - p which is the probability that a randomly selected pen is not defective.<BR>\n" );
document.write( "c(n,x) is the number of ways to select a set of x elements out of a set of n elements where order is not important.\r<BR>\n" );
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document.write( "the formula for c(n,x) is c(n,x) =  n! / (x! * (n-x)!).\r<BR>\n" );
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document.write( "you are looking for the probability that less than or equal to 2 are defective when p = .05.\r<BR>\n" );
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document.write( "when p = .05 and the sample size is 17, then:\r<BR>\n" );
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document.write( "p = .05, q = .95, n = 17, x = 0 to 17, c(n,x) = c(17,x).\r<BR>\n" );
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document.write( "the formula is, once again p(x) = p^x * q^(n-x) * c(n,x).\r<BR>\n" );
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document.write( "less than or equal to 2 defective is 0, 1, or 2 defective only.\r<BR>\n" );
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document.write( "p(0) = .05^0 * .95^17 * c(17,0) = 1 * .4181203352 * 1 = .4181203353<BR>\n" );
document.write( "p(1) = .05^1 * .95^16 * c(17,1) = .05 * .4401266687 * 17 = .3741076684<BR>\n" );
document.write( "p(2) = .05^2 * .95^15 * c(17,2) = .0025 * .4632912302 * 136 = .1575190183\r<BR>\n" );
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document.write( "the sum of p(0) + p(1) + p(3) = .9497470218 which rounds to .95.\r<BR>\n" );
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document.write( "when p = .15 and the sample size is 17, then:\r<BR>\n" );
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document.write( "p = .15, q = .85, n = 17, x = 0 to 17, c(n,x) = c(17,x).\r<BR>\n" );
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document.write( "the formula is, once again p(x) = p^x * q^(n-x) * c(n,x).\r<BR>\n" );
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document.write( "less than or equal to 2 defective is 0, 1, or 2 defective only.\r<BR>\n" );
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document.write( "p(0) = .15^0 * .85^17 * c(17,0) = 1 * .0631134233 * 1 = .0631134233<BR>\n" );
document.write( "p(1) = .15^1 * .85^16 * c(17,1) = .15 * .0742510862 * 17 = .1893402699<BR>\n" );
document.write( "p(2) = .15^2 * .85^15 * c(17,2) = .0225 * .0873542191 * 136 = .2673039104\r<BR>\n" );
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document.write( "the sum of p(0) + p(1) + p(3) = .5197576037 which rounds to .52.\r<BR>\n" );
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document.write( "that's the probability that less than or equal to 2 are defective.\r<BR>\n" );
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document.write( "the probability that more than 2 are defective is therefore 1 - .52 = .48.\r<BR>\n" );
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document.write( "your solution should be.\r<BR>\n" );
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document.write( "the probability that the shipment is accepted if 5% are defective would be .95.\r<BR>\n" );
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document.write( "the probability that the shipment is not accepted if 15% are defective would be .48.\r<BR>\n" );
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document.write( "that's what i get if i analyzed the problem correctly.\r<BR>\n" );
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document.write( "i used excel to calculate all the probabilities.\r<BR>\n" );
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document.write( "the sum of all probabilities is equal to 1 as it should be.\r<BR>\n" );
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document.write( "here's the table for 5% defective.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/100803.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "here's the table for 15% defective.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/100804.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "if you need to show your answer to 3 decimal places, then .950 and .480.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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