document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=199709>Question 199709</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780321458209/algebrahomeworkh>Using and Understanding Mathematics A Quantitative Reasoning Approach</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> find the probability of the given event:  Being dealt 5 cards from a standard 52-card deck and getting a four of a kind, (for example, four aces). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #150082 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=150082\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For any 5 card poker hand, the probability is the number of different ways that hand can be dealt divided by the number of possible 5 card hands in a 52 card deck.  Let's deal with the denominator first.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The number of possible 5 card hands dealt from a 52 card deck is given by the number of combinations of 52 things taken 5 at a time.  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 r\"> at a time is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ _nC_r = \frac{n!}{r!(n - r)!\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So (and you should verify my arithmetic):\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ _{52}C_5 = \frac{52!}{5!(52 - 5)!} = 2,598,960\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now, let's deal with the numerator:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First, the number of ways you can choose which rank your 4 of a kind will be, namely combinations of 13 things taken 1 at a time.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ _{13}C_{1} = \frac{13!}{1!(13 - 1)!} = 13\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(not surprisingly)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then, the number of ways to draw the other three cards of that rank, namely combinations of 4 things taken 4 at a time.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ _{4}C_4 = \frac{4!}{4!(4 - 4)!} = 1\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Once you have decided on the rank, there's only one way to get the other three cards of that rank.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Finally, the number of ways to draw the last card.  Once the 4 of a kind are selected, there are 48 other cards to choose, so the number of combinations of 48 things taken 1 at a time:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ _{48}C_1 = \frac{48!}{1!(48 - 1)!} = 48\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And the total number of ways is the product of each of these counts, so:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ _{13}C_1\cdot_4C_4\cdot_{48}C_1 = 13\cdot1\cdot48 = 624\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Giving us the numerator.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Your probability then is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{_{13}C_1\cdot_4C_4\cdot_{48}C_1}{_{52}C_5} = \frac{624}{2,598,960} \approx 0.000240 = 0.0240%\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To put this in perspective, let's say you sat in on a poker game once per week and played 50 hands each time.  You might expect to be dealt 2 or 3 hands with 4 of a kind in 4 years.  Also, if one of the other guys seems to win a big pot with 4 of a kind every week, then someone is dealing off the bottom of the deck.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "By the way, if you wanted the probability of getting exactly 4 Aces, for example, then reduce the numerator of the above calculation by a factor of 13.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );