document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197443>Question 1197443</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider an experiment involving a random arrangement of the letters of the word <BR>\n" );
document.write( "LEXICOGRAPHY. <BR>\n" );
document.write( "a. Determine the cardinality of the sample space S, i.e., the number of possible <BR>\n" );
document.write( "arrangements of the letters of the word LEXICOGRAPHY.<BR>\n" );
document.write( "Define the random variable B to be the number of vowels in the first six (6) letters of the <BR>\n" );
document.write( "the random arrangement of the letters of the word LEXICOGRAPHY.<BR>\n" );
document.write( "b. What are the possible values of B? Determine the probability mass function of the random variable B. Your solution must be consistent with the sample space in part (a). <BR>\n" );
document.write( "Show complete solutions for the PMF, following the set standard indicated in the <BR>\n" );
document.write( "instructions written above. Express your final answer in a table and using the fractions.<BR>\n" );
document.write( "c. Find the mean and variance of the random variable B.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830760 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=830760\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "There are 12 letters in the word LEXICOGRAPHY.\r<BR>\n" );
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document.write( "There are 12 selections for the first slot, then 11 for the second, 10 for the third, and so on until we get to 1 selection for the last slot.<BR>\n" );
document.write( "12! = 12*11*10*9*...*3*2*1 = 479,001,600<BR>\n" );
document.write( "The exclamation mark indicates a factorial.\r<BR>\n" );
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document.write( "Since there are no repeated letters, we don't have to worry about overcounting.\r<BR>\n" );
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document.write( "Answer: <font color=red>479,001,600</font><BR>\n" );
document.write( "This value is a slightly larger than 479 million.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "I'll use X in place of the variable B.\r<BR>\n" );
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document.write( "X = number of vowels in the first 6 letters of the arrangement of LEXICOGRAPHY\r<BR>\n" );
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document.write( "The four vowels in that word are E, I, O, A. They are recorded in the order they come up in the word LEXICOGRAPHY.\r<BR>\n" );
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document.write( "This means we could have the following vowel count in the first 6 letters <BR>\n" );
document.write( "X = 0 vowels<BR>\n" );
document.write( "X = 1 vowel<BR>\n" );
document.write( "X = 2 vowels<BR>\n" );
document.write( "X = 3 vowels<BR>\n" );
document.write( "X = 4 vowels\r<BR>\n" );
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document.write( "We'll use both the combination formula nCr and the permutation formula nPr to count how many sub-cases there are for X = 0 through X = 4.\r<BR>\n" );
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document.write( "The notation I'll use is C(n,r) for combinations.<BR>\n" );
document.write( "I'll also use P(n,r) for permutations.\r<BR>\n" );
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document.write( "As a refresher, here are the formulas for each<BR>\n" );
document.write( "C(n,r) = (n!)/(r!*(n-r)!)<BR>\n" );
document.write( "P(n,r) = (n!)/((n-r)!)\r<BR>\n" );
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document.write( "An interesting connection is that<BR>\n" );
document.write( "C(n,r) = P(n,r)/(r!)<BR>\n" );
document.write( "which rearranges to <BR>\n" );
document.write( "P(n,r) = (r!)*C(n,r)<BR>\n" );
document.write( "which is handy if you know exactly one of the nPr or nCr values and want to find the other one.\r<BR>\n" );
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document.write( "On spreadsheet software, the commands for nPr and nCr are <font color=blue>PERMUT</font> (not PERMUTE, I don't know why they decided to leave the last E off) and <font color=blue>COMBIN</font> (the CHOOSE function is not what we want since that deals with pulling elements out of a set).\r<BR>\n" );
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document.write( "nPr = PERMUT<BR>\n" );
document.write( "nCr = COMBIN\r<BR>\n" );
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document.write( "Example usage<BR>\n" );
document.write( "Type in <font color=blue>=COMBIN(10,3)</font> to find the value of C(10,3) = 120. <BR>\n" );
document.write( "Use the nCr combination formula mentioned above to verify this result. <BR>\n" );
document.write( "Don't forget about the equal sign up front when typing in the COMBIN function.\r<BR>\n" );
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document.write( "Another example:<BR>\n" );
document.write( "Type in <font color=blue>=PERMUT(9,2)</font> to compute P(9,2); the result of which is 72.<BR>\n" );
document.write( "Don't forget about the equal sign up front.<BR>\n" );
document.write( "Use the nPr formula mentioned above to confirm this result.\r<BR>\n" );
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document.write( "Let's get back to the problem at hand.<BR>\n" );
document.write( "X goes from x = 0 to x = 4.<ul><li>P(4,x) represents the number of ways to pick the x vowels to be somewhere in the first 6 slots.</li><li>C(6,x) has us figure out where those x vowels go in those first six slots.</li><li>P(8,6-x) then computes the number of ways to pick the consonants for the remaining 6-x slots (somewhere in the first 6 slots).</li><li>P(6,6) = 720 is the number of ways to select the remaining 6 letters (slot7,slot8,...,slot12)</li></ul>We'll multiply out those items.\r<BR>\n" );
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document.write( "Here's what the spreadsheet looks like before each item is calculated<BR>\n" );
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document.write( "I'm using LibreOffice spreadsheet, which is free software. Those commands work in Excel and similar spreadsheet programs. \r<BR>\n" );
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document.write( "Calculating each item gets us this<BR>\n" );
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document.write( "Use the command <font color=blue>SUM</font> to add up all the values in column F.<BR>\n" );
document.write( "The command is <font color=blue>=SUM(F2:F6)</font> to sum the values from F2 to F6.<BR>\n" );
document.write( "You should get 479,001,600 as that sum. <BR>\n" );
document.write( "Which if you recall, this is the result of part (a). <BR>\n" );
document.write( "This makes sense because X = 0 through X = 4 represent all of the possible ways to <a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/Partition_of_a_set\">partition</a> the sample space. <BR>\n" );
document.write( "This is what your teacher meant when s/he wrote \"Your solution must be consistent with the sample space in part (a)\". This is a checksum of sorts.\r<BR>\n" );
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document.write( "Column F represents the frequency for cases X = 0 through X = 4.<BR>\n" );
document.write( "For example, there are exactly 14,515,200 permutations in which there aren't any vowels in the first six slots. \r<BR>\n" );
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document.write( "What we'll do is form a new column where we divide values in column F over the grand total 479001600. This is to determine the relative frequency for each case X = 0 through X = 4.\r<BR>\n" );
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document.write( "For example, X = 0 has a frequency of 14,515,200 and relative frequency of 14515200/479001600 which reduces to 1/33. <BR>\n" );
document.write( "Do this for the other values of X as well.\r<BR>\n" );
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document.write( "Here's how to quickly reduce fractions<BR>\n" );
document.write( "In cells G1 and G2 type the numerator and denominator. Both of which need to be integers.<BR>\n" );
document.write( "In cell H1, we'll have this command <font color=blue>=G1/GCD(G1,G2)</font><BR>\n" );
document.write( "In cell H2, we'll have this command <font color=blue>=G2/GCD(G1,G2)</font><BR>\n" );
document.write( "Therefore, the fraction (G1)/(G2) reduces fully to (H1)/(H2).<BR>\n" );
document.write( "GCD = greatest common divisor, it's the same as the GCF\r<BR>\n" );
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document.write( "Answers: <BR>\n" );
document.write( "What are the possible values of B?  <font color=red>0, 1, 2, 3, and 4</font> \r<BR>\n" );
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document.write( "The PMF is this table<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >B</TD><TD >P(B)</TD></tr><tr><TD >0</TD><TD >1/33</TD></tr><tr><TD >1</TD><TD >8/33</TD></tr><tr><TD >2</TD><TD >5/11</TD></tr><tr><TD >3</TD><TD >8/33</TD></tr><tr><TD >4</TD><TD >1/33</TD></tr>\n" );
document.write( "</TABLE>PMF = probability mass function\r<BR>\n" );
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document.write( "Side notes: <BR>\n" );
document.write( "As a check, the P(B) values need to add up to 1.<BR>\n" );
document.write( "5/11 = 15/33\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "I'll use X in place of the variable B.\r<BR>\n" );
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document.write( "Here's the PMF from part (b) earlier.<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr><tr><TD >0</TD><TD >1/33</TD></tr><tr><TD >1</TD><TD >8/33</TD></tr><tr><TD >2</TD><TD >5/11</TD></tr><tr><TD >3</TD><TD >8/33</TD></tr><tr><TD >4</TD><TD >1/33</TD></tr>\n" );
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document.write( "Form a third column labeled X*P(X)<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD></tr><tr><TD >0</TD><TD >1/33</TD><TD >0/33</TD></tr><tr><TD >1</TD><TD >8/33</TD><TD >8/33</TD></tr><tr><TD >2</TD><TD >5/11</TD><TD >30/33</TD></tr><tr><TD >3</TD><TD >8/33</TD><TD >24/33</TD></tr><tr><TD >4</TD><TD >1/33</TD><TD >4/33</TD></tr>\n" );
document.write( "</TABLE>Add up the values in the X*P(X) column and you should get 2 exactly.<BR>\n" );
document.write( "This is the mean of X<BR>\n" );
document.write( "mu is the greek letter that represents the mean.\r<BR>\n" );
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document.write( "Then form a fourth column labeled (X-mu)^2*P(X)<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD><TD >(X-mu)^2*P(X)</TD></tr><tr><TD >0</TD><TD >1/33</TD><TD >0/33</TD><TD >4/33</TD></tr><tr><TD >1</TD><TD >8/33</TD><TD >8/33</TD><TD >8/33</TD></tr><tr><TD >2</TD><TD >5/11</TD><TD >30/33</TD><TD >0/33</TD></tr><tr><TD >3</TD><TD >8/33</TD><TD >24/33</TD><TD >8/33</TD></tr><tr><TD >4</TD><TD >1/33</TD><TD >4/33</TD><TD >4/33</TD></tr>\n" );
document.write( "</TABLE>Add up everything in the fourth column to get the fraction 24/33, which reduces to 8/11. This is the variance of X.<BR>\n" );
document.write( "Side note: Apply the square root to the variance to get the standard deviation.\r<BR>\n" );
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document.write( "Answers: <BR>\n" );
document.write( "Mean = <font color=red>2</font> <BR>\n" );
document.write( "Variance = <font color=red>8/11</font><BR>\n" );
document.write( "Convert the variance to decimal form if your teacher requires it. <BR>\n" );
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