document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=326460>Question 326460</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I would like to see the mechanics of how to figure out the formula for this so I can do this myself.  I have been checking out Permutations at various websites, my efforts have not been fruitful, but I have tried.\r<BR>\n" );
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document.write( "You have a deck of 40 cards of different colors.  The deck contains 3 red cards and 3 blue cards.  The other 34 cards are not red or blue.  The deck is randomly shuffled.\r<BR>\n" );
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document.write( "1) What is the probability of drawing at least 1 red and 1 blue card in the first 6 cards drawn from the top of the deck?\r<BR>\n" );
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document.write( "2) What is the probability of drawing at least 1 red and 1 blue card in the first 10 cards drawn from the top of the deck?\r<BR>\n" );
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document.write( "Thank you.  Ken\r<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 #233748 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=233748\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "This is one of those cases where it will be easier to calculate the probability of utter failure to get what you want and subtract that from 1.  We will need the probability that none of the cards are red plus the probability that none of the cards are blue minus the probability that none of the cards are red or blue.\r<BR>\n" );
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document.write( "Let A = the event that at least one card is Red.  Let B = the event that at least one card is Blue.  Then A or B is the event that at least one card is either red or blue.  ~A means no card is red, ~B means no card is blue, and ~(A or B) means ALL of the cards are something other than red or blue.\r<BR>\n" );
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document.write( "We need to calculate <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1\ -\ \left(P(\neg{A})\ +\ P(\neg{B})\ -\ P\left(\neg(A\,or\,B)\right)\right)\">\r<BR>\n" );
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document.write( "We need to subtract P(~(A or B)) to avoid double counting since P(~A) and P(~B) are NOT mutually exclusive events.\r<BR>\n" );
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document.write( "The probability of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> successes in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> trials where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> is the probability of success on any given trial is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_n(k,p)\ =\ \left(n\cr k\right\)p^k\left(1\,-\,p\right)^{n\,-\,k}\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(n\cr k\right\)\"> is the number of combinations of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> things taken <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> at a time and is calculated by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{n!}{k!(n\,-\,k)!}\">\r<BR>\n" );
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document.write( "First we want the probability that NONE of the first 6 cards are RED:  6 trials, 0 successes, probability of success <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{3}{20}\"> for No REDs:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_6\left(0,\frac{3}{20}\right)\ =\ \left(6\cr 0\right\)\left(\frac{3}{20}\right)^0\left(\frac{17}{20}\right)^6\">\r<BR>\n" );
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document.write( "The arithmetic is simpler than it looks because <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(n\cr 0\right)\ =\ 1\"> always, and anything raised to the zero power is 1, so you really only have to compute <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(\frac{17}{20}\right)^6\">\r<BR>\n" );
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document.write( "Next we want the probability that NONE of the first 6 cards are BLUE:  Again, 6 trials, 0 successes, probability of success <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{3}{20}\"> for No BLUEs:\r<BR>\n" );
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document.write( "It is the same as the number for no reds.\r<BR>\n" );
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document.write( "Last we want the probability that NONE of the first 6 are RED or BLUE: 6 trials, 0 successes, probability of success <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{6}{20}\"> for No REDs or BLUEs:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_6\left(0,\frac{3}{20}\right)\ =\ \left(6\cr 0\right\)\left(\frac{6}{20}\right)^0\left(\frac{14}{20}\right)^6\">\r<BR>\n" );
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document.write( "But all you need to calculate is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(\frac{14}{20}\right)^6\">\r<BR>\n" );
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document.write( "Putting it all together:\r<BR>\n" );
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document.write( "We need to calculate <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1\ -\ \left(P(\neg{A})\ +\ P(\neg{B})\ -\ P\left(\neg(A\,or\,B)\right)\right)\">\r<BR>\n" );
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document.write( "which, in terms of something that will make sense to our calculator is:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1\ -\ \left(\left(\frac{17}{20}\right)^6\ +\ \left(\frac{17}{20}\right)^6\ -\ \left(\frac{14}{20}\right)^6\right)\">\r<BR>\n" );
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document.write( "Just break out the calculator and start punching, or, if you have Excel or something like it handy, open a spreadsheet and set it up to do the calculations  for you.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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