document.write( "Question 1201659: Compute the probability that a 5-card poker hand
\n" ); document.write( "a. is dealt to you that contains all hearts.
\n" ); document.write( "b. hand is dealt to you that contains four Aces.\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #836129 by math_tutor2020(3817) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Answers:

33/66640
1/54145

===============================================================================================\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Work Shown:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Part (a)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "13 hearts out of 52 cards total
\n" ); document.write( "13/52 = probability of getting a heart
\n" ); document.write( "12/51 = probability of getting a second heart
\n" ); document.write( "We do not put the cards back, aka no replacement.
\n" ); document.write( "Subtract 1 from numerator and denominator each time we take out another card.
\n" ); document.write( "11/50 = probability of getting a third heart
\n" ); document.write( "10/49 = probability of getting a fourth heart
\n" ); document.write( "9/48 = probability of getting a fifth heart\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Multiply out the fractions
\n" ); document.write( "(13/52)*(12/51)*(11/50)*(10/49)*(9/48)
\n" ); document.write( "= (13*12*11*10*9)/(52*51*50*49*48)
\n" ); document.write( "= 154440/311875200
\n" ); document.write( "= 33/66640 is the probability of getting 5 hearts in a row. \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Another approach:
\n" ); document.write( "n = 13 hearts
\n" ); document.write( "r = 5 slots to fill
\n" ); document.write( "Use the nCr combination formula. The order doesn't matter in a poker hand.
\n" ); document.write( "n C r = (n!)/(r!(n-r)!)
\n" ); document.write( "13 C 5 = (13!)/(5!*(13-5)!)
\n" ); document.write( "13 C 5 = (13!)/(5!*8!)
\n" ); document.write( "13 C 5 = (13*12*11*10*9*8!)/(5!*8!)
\n" ); document.write( "13 C 5 = (13*12*11*10*9)/(5!)
\n" ); document.write( "13 C 5 = (13*12*11*10*9)/(5*4*3*2*1)
\n" ); document.write( "13 C 5 = 154440/120
\n" ); document.write( "13 C 5 = 1287
\n" ); document.write( "There are 1287 ways to get a hand of nothing but hearts.
\n" ); document.write( "Follow similar steps to find 52C5 = 2598960 different five-card hands.
\n" ); document.write( "1287/2598960 = 33/66640\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-----------------------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Part (b)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "There is 1 way to select the 4 aces where order doesn't matter.
\n" ); document.write( "There are 52-4 = 48 other cards that aren't an ace.
\n" ); document.write( "Ultimately there are 1*48 = 48 five-card hands that consist of 4 aces and some other card.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "This is out of the 2598960 different five-card hands.
\n" ); document.write( "48/2598960 = 1/54145 represents the probability of getting four aces.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-----------------------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "For comparison purposes:
\n" ); document.write( "

\r\n" );
document.write( "33/66640 = 0.000495 = 0.0495% \r\n" );
document.write( " 1/54145 = 0.000018 = 0.0018%\r\n" );
document.write( "

Each decimal value and percentage is approximate.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The event \"getting 5 hearts\" is more likely compared to the event \"getting 4 aces\".
\n" ); document.write( "Both events are really rare on their own.
\n" ); document.write( " \n" ); document.write( "

\n" );