document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1130264>Question 1130264</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose four integers are chosen successively at random between 0 and 12, inclusive. Find the probability that:\r<BR>\n" );
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document.write( "a. they are all different?\r<BR>\n" );
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document.write( "b. not more than 2 are the same? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #747070 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=747070\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I find for this type of problem it is easier to count the number of ways the event(s) of interest<BR>\n" );
document.write( "can occur then divide by the total number of possible outcomes.<br>\r<BR>\n" );
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document.write( "The number of selections of 4 numbers from the set (0, 1, . . ., 12), with replacement, is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+13%5E4+ ALIGN=MIDDLE  ALT=\"+13%5E4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  This will be the denominator in our probability calculation.<br>\r<BR>\n" );
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document.write( "Note: Using notation (nCr  = n!/((n-r)!r!))<br>\r<BR>\n" );
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document.write( "a) There are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4%21%2A13C4+ ALIGN=MIDDLE  ALT=\"+4%21%2A13C4+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways to pick 4 unique numbers from the set of 13 (4! because A,B,C,D<BR>\n" );
document.write( "and C,B,D,A are the same pick but picked in different order).   \r<BR>\n" );
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document.write( "P(no matches) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4%21%2A13C4+%2F+13%5E4+=+17160+%2F+28561+=+highlight%28+1320+%2F+2197+%29+ ALIGN=MIDDLE  ALT=\"+4%21%2A13C4+%2F+13%5E4+=+17160+%2F+28561+=+highlight%28+1320+%2F+2197+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > or about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28+0.6008+%29+ ALIGN=MIDDLE  ALT=\"+highlight%28+0.6008+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "b) P(no more than two match) = P(no matches) + P(exactly two match) = 1320/2197 + P(exactly two match)\r<BR>\n" );
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document.write( "P(exactly two match) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%2813C3%29%284C2%29%2A2%21+%2F+13%5E4+=+3432+%2F+28561+=+264+%2F+2197+ ALIGN=MIDDLE  ALT=\"+%2813C3%29%284C2%29%2A2%21+%2F+13%5E4+=+3432+%2F+28561+=+264+%2F+2197+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  or about <b>0.12016</b><br>\r<BR>\n" );
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document.write( "P(no more than two match) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%281320%2B264%29+%2F+2197++=+highlight%28+1584+%2F+2197+%29+ ALIGN=MIDDLE  ALT=\"+%281320%2B264%29+%2F+2197++=+highlight%28+1584+%2F+2197+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > or about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28+0.7210+%29+ ALIGN=MIDDLE  ALT=\"+highlight%28+0.7210+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <br>\r<BR>\n" );
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document.write( "This is probably the answer your teacher is looking for. <br>\r<BR>\n" );
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document.write( "I'd like to point out: if {A,B,C,D} are any four unique numbers selected from the set {0,1, . . ., 12} then part (b) addresses A,B,C,D (and its permutations) and A,A,B,C  (and its permutations) while the pattern A,A,B,B (and its permutations) are NOT counted.  For example, {7,7,11,11} is NOT counted in part (b) even though technically there are only two matching numbers.  When I interpret \"not more than 2 are the same\" I sometimes think {7,7,11,11} should be included.  <br>\r<BR>\n" );
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document.write( "EDITED:  Fixed the wording on the paragraph about number of selections of 4 numbers, and added this section:<br> \r<BR>\n" );
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document.write( "To include the pattern {A,A,B,B}, add to the above answer: <BR>\n" );
document.write( "  P(AABB) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%2813C2%29%284C2%29+%2F+13%5E4+=+468+%2F+28561+ ALIGN=MIDDLE  ALT=\"+%2813C2%29%284C2%29+%2F+13%5E4+=+468+%2F+28561+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  to part (b).\r<BR>\n" );
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document.write( " </SPAN>\n" );
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