document.write( "Question 1035506: you are dealt a 5 card hand. whats is the probability that the hand you have is a straight (aces are high)? \n" ); document.write( "
Algebra.Com's Answer #650184 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We will find the number of straights including straight flushes\r\n" );
document.write( "and then subtract the number of straight flushes, in case you\r\n" );
document.write( "don't want to include those:\r\n" );
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document.write( "We choose the denominations any of these 10 ways:\r\n" );
document.write( "A,2,3,4,5\r\n" );
document.write( "2,3,4,5,6\r\n" );
document.write( "3,4,5,6,7\r\n" );
document.write( "4,5,6,7,8\r\n" );
document.write( "5,6,7,8,9\r\n" );
document.write( "6,7,8,9,10\r\n" );
document.write( "7,8,9,10,J\r\n" );
document.write( "8,9,10,J,Q\r\n" );
document.write( "9,10,J,Q,K\r\n" );
document.write( "10,J,Q,K,A\r\n" );
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document.write( "For each of those ways we choose the suit for\r\n" );
document.write( "the lowest card in 4 ways. That's 10*4 ways\r\n" );
document.write( "For each of those ways we choose the suit for\r\n" );
document.write( "the next to lowest card in 4 ways. That's 10*4*4\r\n" );
document.write( "For each of those ways we choose the suit for\r\n" );
document.write( "the middle card in 4 ways. That's 10*4*4*4\r\n" );
document.write( "For each of those ways we choose the suit for\r\n" );
document.write( "the next to highest card in 4 ways. That's 10*4*4*4*4\r\n" );
document.write( "For each of those ways we choose the suit for\r\n" );
document.write( "the highest card in 4 ways. That's 10*4*4*4*4*4\r\n" );
document.write( "ways or 10*4^5 = 10240\r\n" );
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document.write( "We now find the number of straight flushes \r\n" );
document.write( "(including Royal flushes).\r\n" );
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document.write( "We choose the denominations any of these 10 ways:\r\n" );
document.write( "A,2,3,4,5\r\n" );
document.write( "2,3,4,5,6\r\n" );
document.write( "3,4,5,6,7\r\n" );
document.write( "4,5,6,7,8\r\n" );
document.write( "5,6,7,8,9\r\n" );
document.write( "6,7,8,9,10\r\n" );
document.write( "7,8,9,10,J\r\n" );
document.write( "8,9,10,J,Q\r\n" );
document.write( "9,10,J,Q,K\r\n" );
document.write( "10,J,Q,K,A\r\n" );
document.write( "We can choose the 1 suit they must all have to be \r\n" );
document.write( "a straight flush in 4 ways.\r\n" );
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document.write( "That's 10*4 or 40 straight flushes.\r\n" );
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document.write( "So the number of straights is 10240-40 = 10200\r\n" );
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document.write( "The number of 5-card poker hands is\r\n" );
document.write( "52 cards choose 5 = 52C5 = 2598960.\r\n" );
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document.write( "So the probability of being dealt a straight that\r\n" );
document.write( "is not a straight flush is \r\n" );
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document.write( "10200/2598960 = 5/1274 (dividing top and bottom by 2040).\r\n" );
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document.write( "That's about 0.003924646781789638932496075353218210361067503924646781789\r\n" );
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document.write( "If you want the probability including the straight\r\n" );
document.write( "flushes, then it's\r\n" );
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document.write( "10240/2598960 = 128/32487 (dividing top and bottom by 80).\r\n" );
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document.write( "That's about 0.003940037553482931634192138393819066087973650998861082894…\r\n" );
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document.write( "[They are the same rounded to 4 decimal places. 0.0039.  That's \r\n" );
document.write( "because straight flushes are so rare].\r\n" );
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document.write( "Edwin
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