document.write( "Question 1204454: A standard deck of cards contains 52 cards. These 52 cards are arranged in a circle, at random. Find the expected number of pairs of adjacent cards that are both Aces. \n" ); document.write( "
Algebra.Com's Answer #840774 by Edwin McCravy(20064) $\"\"$ $\"About$
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document.write( "Even though all 52 cards of a deck are distinguishable, we can simplify matters by\r\n" );
document.write( "considering all 4 aces as indistinguishable from each other, and the 48 non-aces also as\r\n" );
document.write( "indistinguishable from each other.\r\n" );
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document.write( "I will first justify this assumption of indistinguishability by what is meant by\r\n" );
document.write( "\"expectation\".\r\n" );
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document.write( "The expectation in this case is the average number of adjacent aces per shuffle.\r\n" );
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document.write( "Suppose we performed the experiment many times of shuffling a deck of cards and after\r\n" );
document.write( "each shuffle, we counted and kept a running average of the number of adjacent pairs of\r\n" );
document.write( "aces per shuffle. The expectation of the number of adjacent pairs of aces is the number\r\n" );
document.write( "that should be close to the running average of adjacent pairs of aces per shuffle.\r\n" );
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document.write( "My justification is by thinking of the way we would perform each experiment of going\r\n" );
document.write( "through the cards to count the number of adjacent pairs of aces. As we would go through\r\n" );
document.write( "the deck of cards looking for adjacent pairs of aces, we would pay no attention to the\r\n" );
document.write( "suit of any of the cards, and we would also pay no attention to the rank of any\r\n" );
document.write( "of the cards except merely to determine whether it is an ace or not.\r\n" );
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document.write( "So I will go by my assumption. It is the same as if we had a deck of 4 white cards and\r\n" );
document.write( "48 black cards and we wanted to find the expected number of adjacent pairs of white\r\n" );
document.write( "cards per shuffle. But I go back to aces and non-aces.\r\n" );
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document.write( "We begin by laying out the 48 \"indistinguishable\" non-aces in a circle, leaving a space\r\n" );
document.write( "between each non-ace.  We will now place the 4 \"indistinguishable\" aces in some of those\r\n" );
document.write( "48 spaces between cards.  Remember, we can put more than 1 ace in a single space between\r\n" );
document.write( "two non-aces.\r\n" );
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document.write( "We can insert the 4 aces in some of 48 available spaces between non-aces in the\r\n" );
document.write( "following cases, which correspond to the 5 ways to write 4 as a sum of positive\r\n" );
document.write( "integers or as a single positive integer:\r\n" );
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document.write( "1+1+1+1 = 2+1+1 = 2+2 = 3+1 = 4\r\n" );
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document.write( "Case 1.  4 separated aces (no adjacent pairs) [That's 0 adjacent pairs]\r\n" );
document.write( "Case 2.  1 adjacent pair of aces and 2 aces separated. [That's 1 adjacent pair]\r\n" );
document.write( "Case 3.  2 adjacent pairs of aces.\r\n" );
document.write( "Case 4.  3 aces together and 1 ace separated. [That's also 2 pairs of adjacent aces]\r\n" );
document.write( "Case 5.  All 4 aces together. [That's 3 pairs of adjacent aces]\r\n" );
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document.write( "Case 1. There are 48C4=194580 ways to place the 4 aces, no two together.\r\n" );
document.write( "Case 2. There are 48 ways to place the adjacent pair and 47C2=1081 ways to place the\r\n" );
document.write( "        separated two.  That's (48)(1081)=51888 ways.\r\n" );
document.write( "Case 3. There are 48C2=1128 ways to place the two pairs of adjacent aces.\r\n" );
document.write( "Case 4. There are 48 ways to place the 3 aces together and then 47 ways to place the\r\n" );
document.write( "        separated ace.  Thats (48)(47)=2256 ways.\r\n" );
document.write( "Case 5. There are 48 ways to place all 4 aces together.\r\n" );
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document.write( "The total number of ways to place the aces is\r\n" );
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document.write( "194580 + 51888 + 1128 + 2256 + 48 = 249900\r\n" );
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document.write( "The probability that there are 0 (no) adjacent pairs = 194580/249900 = 3243/4165\r\n" );
document.write( "The probability that there is exactly 1 adjacent pair = 51888/249900 = 4324/20825\r\n" );
document.write( "The probability that there are exactly 2 adjacent pairs = (1128+2256)/249900 = 282/20825\r\n" );
document.write( "The probability that there are exactly 3 adjacent pairs = 48/249900 = 4/20825\r\n" );
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document.write( "The expected number of pairs of aces together is\r\n" );
document.write( "0(3243/4165) + 1(4324/20825) + 2(282/20825) + 3(4/20825) = 4/17  <--answer\r\n" );
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document.write( "A recap of what this 4/17 now means:\r\n" );
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document.write( "Out of every 17 shuffles you expect to average counting about 4 pairs of adjacent aces. \r\n" );
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document.write( "Edwin
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