document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194848>Question 1194848</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a 5-card poker hand, find the probability that the hand contains 3 spades. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #827133 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=827133\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "There are 2,598,960 different five-card hands possible (whether they have 3 spades or not).\r<BR>\n" );
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document.write( "The scratch work for this is shown here<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "52 C 5 = (52!)/(5!*(52-5)!)<BR>\n" );
document.write( "52 C 5 = (52!)/(5!*47!)<BR>\n" );
document.write( "52 C 5 = (52*51*50*49*48*47!)/(5!*47!)<BR>\n" );
document.write( "52 C 5 = (52*51*50*49*48)/(5!)<BR>\n" );
document.write( "52 C 5 = (52*51*50*49*48)/(5*4*3*2*1)<BR>\n" );
document.write( "52 C 5 = (311875200)/(120)<BR>\n" );
document.write( "52 C 5 = 2,598,960<BR>\n" );
document.write( "We use the nCr combination formula because order doesn't matter in a card hand.<BR>\n" );
document.write( "A hand like {Q, 2, 3, J, K} is the same as {2, 3, J, Q, K}\r<BR>\n" );
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document.write( "There are 13 spades and we want exactly 3 spades<BR>\n" );
document.write( "Let's plug in n = 13 and r = 3<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "13 C 3 = (13!)/(3!*(13-3)!)<BR>\n" );
document.write( "13 C 3 = (13!)/(3!*10!)<BR>\n" );
document.write( "13 C 3 = (13*12*11*10!)/(3!*10!)<BR>\n" );
document.write( "13 C 3 = (13*12*11)/(3!)<BR>\n" );
document.write( "13 C 3 = (13*12*11)/(3*2*1)<BR>\n" );
document.write( "13 C 3 = (1716)/(6)<BR>\n" );
document.write( "13 C 3 = 286<BR>\n" );
document.write( "There are 286 ways to pick those three spades where order doesn't matter. <BR>\n" );
document.write( "Then there are 39C2 = 741 ways to pick the other two non-spade cards. I'll skip the steps in showing how to get the 741, but you'd follow the templates above to show your teacher how you got 741.\r<BR>\n" );
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document.write( "We have 286*741 = 211,926 ways to pick a five-card hand that has exactly three spades and two other non-spade cards.\r<BR>\n" );
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document.write( "Divide the results we found<BR>\n" );
document.write( "A = number of hands that have 3 spades + 2 others<BR>\n" );
document.write( "B = number of five card hands total<BR>\n" );
document.write( "A = 211,926<BR>\n" );
document.write( "B = 2,598,960<BR>\n" );
document.write( "A/B = probability of getting exactly 3 spades and 2 other non-spades<BR>\n" );
document.write( "A/B = (211,926)/(2,598,960)<BR>\n" );
document.write( "A/B = 0.081543\r<BR>\n" );
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document.write( "Normally I would stick to the fraction form, but the integer values are quite large. <BR>\n" );
document.write( "It might make more intuitive sense to go for the approximate decimal form. <BR>\n" );
document.write( "Of course it's best to follow whatever instructions your teacher provides when it comes to recording the final answer.\r<BR>\n" );
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document.write( "Answer as a fraction: (211,926)/(2,598,960)<BR>\n" );
document.write( "Answer in decimal form: 0.081543 which is approximate<BR>\n" );
document.write( "Answer in percent form: 8.1543% which is also approximate<BR>\n" );
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