document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203217>Question 1203217</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please show steps of how to solve the question. \r<BR>\n" );
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document.write( "Calculate the probability (as a percent) of the following 5-card poker hand:\r<BR>\n" );
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document.write( "   A full house (three cards of one rank and two of another rank)\r<BR>\n" );
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document.write( "For example: 9♠, 9♥, 9♦, 2♣, 2♥\r<BR>\n" );
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document.write( "The Probability is:    %\r<BR>\n" );
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document.write( "(round your percent answer to three decimal places) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838605 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=838605\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red size=4>0.144%</font> (approximate)\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "Use the nCr combination formula to determine how many five card hands are possible. Order does not matter.\r<BR>\n" );
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document.write( "n = 52 cards<BR>\n" );
document.write( "r = 5 selections<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 = 2598960\r<BR>\n" );
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document.write( "There are 2,598,960 ways to select five cards in any order.\r<BR>\n" );
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document.write( "Many calculators have a button labeled \"nCr\" to make such calculations quick and painless.\r<BR>\n" );
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document.write( "Here is one online calculator<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.gigacalculator.com/calculators/combination-calculator.php\">https://www.gigacalculator.com/calculators/combination-calculator.php</a>\r<BR>\n" );
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document.write( "The next step is to determine how many full house hands are possible.\r<BR>\n" );
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document.write( "Focus on the example trio 9♠, 9♥, 9♦<BR>\n" );
document.write( "(ie 9 of spades, 9 of hearts, 9 of diamonds)<BR>\n" );
document.write( "There's 1 card left out (9 of clubs). We do this to avoid getting 4 of a kind.<BR>\n" );
document.write( "There are 4 ways to pick the unlucky card that gets left out.\r<BR>\n" );
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document.write( "Also there are 13 choices for the rank of this trio.<BR>\n" );
document.write( "That gives 13*4 = 52 ways to form the trio.\r<BR>\n" );
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document.write( "For the duo (2♣, 2♥) we have 2 cards left out.<BR>\n" );
document.write( "There are 4C2 = 6 ways to determine the suits for this duo. And there are 12 remaining ranks to pick from.<BR>\n" );
document.write( "We have 6*12 = 72 ways to select the duo.\r<BR>\n" );
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document.write( "We found there are...<ul><li>52 ways to form the trio</li><li>72 ways to form the duo (after the trio is formed)</li></ul>It yields 52*72 = 3744 different full house hands.<BR>\n" );
document.write( "Again, order does not matter.\r<BR>\n" );
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document.write( "-------------------------\r<BR>\n" );
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document.write( "In summary there are:<ul><li>3744 ways to get a full house</li><li>2,598,960 ways to select any five cards</li></ul>Order does not matter.\r<BR>\n" );
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document.write( "Divide the values<BR>\n" );
document.write( "3744/(2,598,960) = 0.0014405762305<BR>\n" );
document.write( "This value is approximate.\r<BR>\n" );
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document.write( "Multiply by 100 to convert to a percent.<BR>\n" );
document.write( "This moves the decimal point 2 spots to the right.\r<BR>\n" );
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document.write( "0.0014405762305 = 0.14405762305%\r<BR>\n" );
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document.write( "That rounds to <font color=red size=4>0.144%</font> when rounding to 3 decimal places.\r<BR>\n" );
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document.write( "Further Reading:<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Poker_probability\">https://en.wikipedia.org/wiki/Poker_probability</a><BR>\n" );
document.write( "and <BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://math.stackexchange.com/questions/808314/probability-of-getting-a-full-house\">https://math.stackexchange.com/questions/808314/probability-of-getting-a-full-house</a><BR>\n" );
document.write( "and <BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://socratic.org/questions/592b200311ef6b2839d30feb\">https://socratic.org/questions/592b200311ef6b2839d30feb</a><BR>\n" );
document.write( "and <BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://math.stackexchange.com/questions/4153143/how-many-ways-are-there-to-get-a-full-house\">https://math.stackexchange.com/questions/4153143/how-many-ways-are-there-to-get-a-full-house</a><BR>\n" );
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