document.write( "Question 1187199: Compute the probability that a five-card poker hand is dealt to you that contains 2 Aces.
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Algebra.Com's Answer #818142 by math_tutor2020(3817)\"\" \"About 
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\n" ); document.write( "With poker hands, the order of the cards does not matter. That means we use the nCr combination formula\r
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\n" ); document.write( "\n" ); document.write( "nCr = (n!)/(r!*(n-r)!)\r
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\n" ); document.write( "\n" ); document.write( "First consider the aces. We have n = 4 aces to choose from and have r = 2 slots to fill for those aces. Order doesn't matter.\r
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\n" ); document.write( "\n" ); document.write( "nCr = (n!)/(r!*(n-r)!)
\n" ); document.write( "4C2 = (4!)/(2!*(4-2)!)
\n" ); document.write( "4C2 = (4!)/(2!*2!)
\n" ); document.write( "4C2 = (4*3*2*1)/(2*1*2*1)
\n" ); document.write( "4C2 = 24/4
\n" ); document.write( "4C2 = 6
\n" ); document.write( "There are 6 ways to choose the two aces from a pool of four.\r
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\n" ); document.write( "\n" ); document.write( "Since the list is short, we can list out all the possible combos
  1. AC,AD
  2. AC,AH
  3. AC,AS
  4. AD,AH
  5. AD,AS
  6. AH,AS
A = ace
\n" ); document.write( "C = clubs
\n" ); document.write( "D = diamonds
\n" ); document.write( "H = hearts
\n" ); document.write( "S = spades
\n" ); document.write( "Something like \"AC\" means \"ace of clubs\"
\n" ); document.write( "Order doesn't matter. Something like AC,AD is the same as AD,AC.\r
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\n" ); document.write( "\n" ); document.write( "Now we consider the three other cards that aren't aces. There are 52 cards total in a deck. Kicking out the aces leaves us with 52-4 = 48 left over.\r
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\n" ); document.write( "\n" ); document.write( "We'll use the nCr formula again to figure out how many ways to pick the three non-ace cards.
\n" ); document.write( "nCr = (n!)/(r!*(n-r)!)
\n" ); document.write( "48C3 = (48!)/(3!*(48-3)!)
\n" ); document.write( "48C3 = (48*47*46*45!)/(3!*45!)
\n" ); document.write( "48C3 = (48*47*46)/(3!)
\n" ); document.write( "48C3 = (48*47*46)/(3*2*1)
\n" ); document.write( "48C3 = (103,776)/6
\n" ); document.write( "48C3 = 17,296
\n" ); document.write( "There are 17,296 ways to pick the three non-aces.\r
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\n" ); document.write( "\n" ); document.write( "To summarize, we found that there are
  • 6 ways to pick the two aces
  • 17,296 ways to pick the three non-aces
Overall, we have 6*17,296 = 103,776 ways to form the five-card hand we're after.\r
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\n" ); document.write( "\n" ); document.write( "This is out of 52C5 = 2,598,960 ways to form any five-card hand (without any restrictions). The steps to computing this are similar to the other set of steps above.\r
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\n" ); document.write( "\n" ); document.write( "The probability we want to compute is therefore:
\n" ); document.write( "(103,776)/(2,598,960) = 0.0399298\r
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\n" ); document.write( "\n" ); document.write( "Answer: Approximately 0.0399298
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