document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208472>Question 1208472</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The chance that a U.S. police chief believes the death penalty “significantly reduces the number of homicides” is 1 in 4. If a random sample of 8 police chiefs is selected, find the probability that at most 3 believe that the death penalty significantly reduces the number of homicides. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846910 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=846910\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer:  <font color=red>0.88618</font>  (approximate)\r<BR>\n" );
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document.write( "<font size=4>Quick way to find the answer using technology</font>\r<BR>\n" );
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document.write( "On a TI83 or similar, you would use the <a  rel=nofollow HREF=\"https://www.statology.org/binomial-probabilities-ti-84-calculator/\">BinomCDF</a> command. <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 = number of trials = 8\r\n" );
document.write( "p = probability of success = 0.25\r\n" );
document.write( "k = number of successes = 3</PRE>Type in <font color=red>BinomCDF(8,0.25,3)</font> to get the approximate result <font color=red>0.88618</font><BR>\n" );
document.write( "The answer will vary depending how you round it.\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</a> in GeoGebra. Select \"binomial\" from the dropdown menu. Type n = 8 and p = 0.25; the goal is to calculate <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large P( 0 \le \text{x} \le 3 )\"></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>=BinomDist(3,8,0.25,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>BinomialDist(8,0.25,3,true)</font></li></ul>Refer to the help manual for more information.\r<BR>\n" );
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document.write( "<font size=4>The slightly longer method</font>\r<BR>\n" );
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document.write( "The binomial probability formula is<BR>\n" );
document.write( "B(x) = (nCx)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "where,<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 = number of trials\r\n" );
document.write( "p = probability of success\r\n" );
document.write( "x = number of successes</PRE>The nCx refers to the nCr combination formula. These values are found in Pascal's Triangle. 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( "Like with many things in math, there are many options to calculate nCr.\r<BR>\n" );
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document.write( "Let's calculate the probability of exactly 0 police chiefs believe that the death penalty significantly reduces the number of homicides.<BR>\n" );
document.write( "B(x) = (nCx)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "B(x) = (8Cx)*(0.25^x)*(1-0.25)^(8-x)<BR>\n" );
document.write( "B(0) = (8C0)*(0.25^0)*(1-0.25)^(8-0)<BR>\n" );
document.write( "B(0) = (1)*(0.25^0)*(1-0.25)^(8-0)<BR>\n" );
document.write( "B(0) = 0.10011292<BR>\n" );
document.write( "This value is approximate.<BR>\n" );
document.write( "It is possible to calculate by hand, given a very long time, but I recommend a calculator. \r<BR>\n" );
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document.write( "Repeat the process to find these other values<BR>\n" );
document.write( "B(1) = 0.26696777<BR>\n" );
document.write( "B(2) = 0.3114624<BR>\n" );
document.write( "B(3) = 0.2076416<BR>\n" );
document.write( "I'll skip showing the steps for these. \r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "B(0)+B(1)+B(2)+B(3)<BR>\n" );
document.write( "= 0.10011292+0.26696777+0.3114624+0.2076416<BR>\n" );
document.write( "= 0.88618469<BR>\n" );
document.write( "= <font color=red>0.88618</font><BR>\n" );
document.write( "The answer will vary depending how you round it.\r<BR>\n" );
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document.write( "More practice with the Binomial Distribution is found on <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1207647.html\">this page</a><BR>\n" );
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