document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208645>Question 1208645</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An integrated circuit manufacturer produces wafers that contain<BR>\n" );
document.write( "20 chips. Each chip has a probability of 0.085 of not placed correctly on the wafer.<BR>\n" );
document.write( "A) Find the probability that a wafer contains at least 3<BR>\n" );
document.write( "incorrectly place chips. (Ans = 0.2390)<BR>\n" );
document.write( "b) What is the probability that a wafer no more than one<BR>\n" );
document.write( "incorrectly place chips on a wafer? (Ans = 0.4836)<BR>\n" );
document.write( "c) What is the expected number of incorrectly placed chips on a<BR>\n" );
document.write( "wafer? (Ans = 1.7) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847029 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=847029\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Please do not type in all upper case. <BR>\n" );
document.write( "I have edited your question to a more readable format.\r<BR>\n" );
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document.write( "We can use the Binomial probability distribution because:<ul><li>There are two outcomes per trial. Either a chip is placed correctly or it is not.</li><li>Each trial is independent.</li><li>The probability of not getting placed correctly is the same for each trial.</li></ul>------------------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Part A\r<BR>\n" );
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document.write( "n = 20 = sample size = number of chips<BR>\n" );
document.write( "p = 0.085 = probability of a chip being incorrectly placed<BR>\n" );
document.write( "x = number of chips incorrectly placed<BR>\n" );
document.write( "x is a whole number that ranges from x = 0 to x = 20, i.e. it's an integer from the set {0,1,2,...,19,20}\r<BR>\n" );
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document.write( "B(x) = binomial probability of exactly x chips being incorrectly placed<BR>\n" );
document.write( "B(x) = (nCx)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "The nCx refers to the <a  rel=nofollow HREF=\"https://www.calculatorsoup.com/calculators/discretemathematics/combinations.php\">nCr combination formula</a>. These values are found in <a  rel=nofollow HREF=\"https://www.mathsisfun.com/pascals-triangle.html\">Pascal's Triangle</a>. A quick way to calculate the nCr values is to use the <a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/office/combin-function-12a3f276-0a21-423a-8de6-06990aaf638a\">Combin</a> function in a spreadsheet. Or you can use a <a  rel=nofollow HREF=\"https://education.ti.com/en/customer-support/knowledge-base/ti-83-84-plus-family/product-usage/34718\">TI Calculator</a>. <BR>\n" );
document.write( "There are many options to calculate nCr.\r<BR>\n" );
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document.write( "Let's find the probability of exactly x = 0 chips that are incorrectly placed.<BR>\n" );
document.write( "B(x) = (nCx)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "B(x) = (20Cx)*(0.085^x)*(1-0.085)^(20-x)<BR>\n" );
document.write( "B(0) = (20C0)*(0.085^0)*(1-0.085)^(20-0)<BR>\n" );
document.write( "B(0) = (1)*(0.085^0)*(1-0.085)^(20-0)<BR>\n" );
document.write( "B(0) = <font color=blue>0.16920839 approximately</font>\r<BR>\n" );
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document.write( "Now let's find the probability of exactly x = 1 chip being incorrectly placed.<BR>\n" );
document.write( "B(x) = (20Cx)*(0.085^x)*(1-0.085)^(20-x)<BR>\n" );
document.write( "B(1) = (20C1)*(0.085^1)*(1-0.085)^(20-1)<BR>\n" );
document.write( "B(1) = (20)*(0.085^1)*(1-0.085)^(20-1)<BR>\n" );
document.write( "B(1) = <font color=blue>0.31437624 approximately</font>\r<BR>\n" );
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document.write( "Repeat similar steps to find that B(2) = <font color=blue>0.27744132 approximately</font> which is the probability of having exactly 2 chips incorrectly placed.\r<BR>\n" );
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document.write( "Then we could say,<BR>\n" );
document.write( "<font size=5>(</font> <font color=blue>B(0)+B(1)+B(2)</font> <font size=5>)</font> + <font size=5>(</font> <font color=red>B(3)+B(4)+...+B(19)+B(20)</font> <font size=5>)</font> = 1<BR>\n" );
document.write( "<font color=red>B(3)+B(4)+...+B(19)+B(20)</font> = 1 - <font size=5>(</font> <font color=blue>B(0)+B(1)+B(2)</font> <font size=5>)</font><BR>\n" );
document.write( "B(3)+B(4)+...+B(19)+B(20) = 1 - ( <font color=blue>0.16920839+0.31437624+0.27744132</font> )<BR>\n" );
document.write( "B(3)+B(4)+...+B(19)+B(20) = 1 - 0.76102595 <BR>\n" );
document.write( "B(3)+B(4)+...+B(19)+B(20) = 0.23897405<BR>\n" );
document.write( "B(3)+B(4)+...+B(19)+B(20) = <font color=red>0.2390</font> which is the answer that your textbook mentioned. \r<BR>\n" );
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document.write( "If you want to use technology to quickly find the answer, then you can use a TI83 or similar. <BR>\n" );
document.write( "The command you'll use is called <a  rel=nofollow HREF=\"https://www.statology.org/binomial-probabilities-ti-84-calculator/\">BinomCDF</a><BR>\n" );
document.write( "The order of inputs is: n, p, k<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">n = 20\r\n" );
document.write( "p = 0.085\r\n" );
document.write( "k = 2</PRE>Type in <font color=red>1-BinomCDF(20,0.085,2)</font> to get the approximate result <font color=red>0.2390</font> when rounding to 4 decimal places.<BR>\n" );
document.write( "You can round manually or you can use the Round command in the TI83.<BR>\n" );
document.write( "Note that the BinomCDF(20,0.085,2) portion computes the sum B(0)+B(1)+B(2)\r<BR>\n" );
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document.write( "Here are some alternative technology options.<ul><li>Search out \"binomial CDF calculator\". <a  rel=nofollow HREF=\"https://stattrek.com/online-calculator/binomial\">This page</a> and <a  rel=nofollow HREF=\"https://www.gigacalculator.com/calculators/binomial-probability-calculator.php\">this page</a> are two of many results. Feel free to explore your favorite.</li><li>Use the <a  rel=nofollow HREF=\"https://geogebra.github.io/docs/manual/en/Probability_Calculator/\">Probability Calculator in GeoGebra</a>. Select \"binomial\" from the dropdown menu. Type n = 20 and p = 0.085; the goal is to calculate <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large P( 3 \le \text{x} \le 20 )\"></li><li>Use the spreadsheet command called <a  rel=nofollow HREF=\"https://support.google.com/docs/answer/3093987?hl=en\">BinomDist</a>. The input would be <font color=red>=1-BinomDist(2,20,0.085,true)</font></li><li>Use the <a  rel=nofollow HREF=\"https://geogebra.github.io/docs/manual/en/commands/BinomialDist/\">BinomialDist</a> command in GeoGebra. Note \"binomial\" instead of \"binom\". The input would be <font color=red>1-BinomialDist(20,0.085,2,true)</font></li></ul>Refer to the software's help manual for more information.\r<BR>\n" );
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document.write( "Part B\r<BR>\n" );
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document.write( "Refer to the previous part. <BR>\n" );
document.write( "We calculated these approximate values<BR>\n" );
document.write( "B(0) = 0.16920839<BR>\n" );
document.write( "B(1) = 0.31437624<BR>\n" );
document.write( "They add up to 0.48358463 which rounds to <font color=red>0.4836</font><BR>\n" );
document.write( "This is the approximate probability of having at most 1 chip being incorrectly placed (i.e. 1 or fewer).\r<BR>\n" );
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document.write( "If you're using technology, then...<ul><li><font color=red>BinomCDF(20,0.085,1)</font> is the input on a TI83 or similar.</li><li><font color=red>=BinomDist(1,20,0.085,true)</font> is the input for a spreadsheet. Don't forget about the equal sign up front.</li><li><font color=red>BinomialDist(20,0.085,1,true)</font> is the input for GeoGebra (either CAS or normal mode). Or you can use the <a  rel=nofollow HREF=\"https://geogebra.github.io/docs/manual/en/Probability_Calculator/\">Probability Calculator in GeoGebra</a>. </li></ul><BR>\n" );
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document.write( "Part C\r<BR>\n" );
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document.write( "n = 20<BR>\n" );
document.write( "p = 0.085<BR>\n" );
document.write( "mean = expected value = n*p = 20*0.085 = <font color=red>1.7</font><BR>\n" );
document.write( "This result is exact and hasn't been rounded.\r<BR>\n" );
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document.write( "More practice with Binomial distributions is found <a  rel=nofollow HREF=\"https://www.algebra.com/statistics/Binomial-probability/Binomial-probability.faq.question.1208472.html\">here</a><BR>\n" );
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