document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=891951>Question 891951</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If one person picks two letters from the alphabet at random (e.g. AB, CG, FO, ZZ [repetition is allowed]), and another person does likewise, what is the probability of the two people choosing the exact same letters?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #540185 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=540185\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "let's try a smaller set to see if we can come up with an algorithm that will give us the answer.\r<BR>\n" );
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document.write( "assume the alphabet only contained 3 letters (ABC)<BR>\n" );
document.write( "the number of possible sets of 2 that you can get from this alphabet is equal to C(3,2) = 3.\r<BR>\n" );
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document.write( "Those 2 letter sets are:\r<BR>\n" );
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document.write( "AB<BR>\n" );
document.write( "AC<BR>\n" );
document.write( "BC\r<BR>\n" );
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document.write( "each person has the potential to get any of these 3 sets.\r<BR>\n" );
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document.write( "the total possible pairings from the 2 sets are therefore 3 * 3 = 9\r<BR>\n" );
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document.write( "those possible pairings are:\r<BR>\n" );
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document.write( "AB AB ***<BR>\n" );
document.write( "AB AC<BR>\n" );
document.write( "AB BC<BR>\n" );
document.write( "AC AB<BR>\n" );
document.write( "AC AC ***<BR>\n" );
document.write( "AC BC<BR>\n" );
document.write( "BC AB<BR>\n" );
document.write( "BC AC<BR>\n" );
document.write( "BC BC ***\r<BR>\n" );
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document.write( "if the first person chooses AB, then the probability that the second person will choose AB is 1/9.\r<BR>\n" );
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document.write( "if the first person chooses AC, then the probability that the second person will choose AC is 1/9.\r<BR>\n" );
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document.write( "if the first person chooses BC, then the probability that the second person will choose BC is 1/9.\r<BR>\n" );
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document.write( "the total probability appears to be 3/9 which is equal to 1/3.\r<BR>\n" );
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document.write( "it appears that the probability that they will match will be equal to C(3,2) / (C(3,2)^2 which is equal to 1 / C(3,2)\r<BR>\n" );
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document.write( "1 / (C(3,2) is equal to 1/3\r<BR>\n" );
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document.write( "let's see if this works with 4.\r<BR>\n" );
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document.write( "you have 4 letters in the alphabet.\r<BR>\n" );
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document.write( "they are ABCD\r<BR>\n" );
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document.write( "you are picking 2 out of the 4.\r<BR>\n" );
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document.write( "the possible combinations are C(4,2) = (4*3)/(1*2) = 6\r<BR>\n" );
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document.write( "those possibilities are:\r<BR>\n" );
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document.write( "AB<BR>\n" );
document.write( "AC<BR>\n" );
document.write( "AD<BR>\n" );
document.write( "BC<BR>\n" );
document.write( "BD<BR>\n" );
document.write( "CD\r<BR>\n" );
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document.write( "that' 6 possible sets of 2 for each person.<BR>\n" );
document.write( "the possible pairings should be 6 * 6 = 36\r<BR>\n" );
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document.write( "those possible pairings are:\r<BR>\n" );
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document.write( "AB AB *****<BR>\n" );
document.write( "AB AC<BR>\n" );
document.write( "AB AD<BR>\n" );
document.write( "AB BC<BR>\n" );
document.write( "AB BD<BR>\n" );
document.write( "AB CD\r<BR>\n" );
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document.write( "AC AB<BR>\n" );
document.write( "AC AC *****<BR>\n" );
document.write( "AC AD<BR>\n" );
document.write( "AC BC<BR>\n" );
document.write( "AC BD<BR>\n" );
document.write( "AC CD\r<BR>\n" );
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document.write( "AD AB<BR>\n" );
document.write( "AD AC<BR>\n" );
document.write( "AD AD *****<BR>\n" );
document.write( "AD BC<BR>\n" );
document.write( "AD BD<BR>\n" );
document.write( "AD CD\r<BR>\n" );
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document.write( "BC AB<BR>\n" );
document.write( "BC AC<BR>\n" );
document.write( "BC AD<BR>\n" );
document.write( "BC BC *****<BR>\n" );
document.write( "BC BD<BR>\n" );
document.write( "BC CD\r<BR>\n" );
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document.write( "BD AB<BR>\n" );
document.write( "BD AC<BR>\n" );
document.write( "BD AD<BR>\n" );
document.write( "BD BC<BR>\n" );
document.write( "BD BD *****<BR>\n" );
document.write( "BD CD\r<BR>\n" );
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document.write( "CD AB<BR>\n" );
document.write( "CD AC<BR>\n" );
document.write( "CD AD<BR>\n" );
document.write( "CD BC<BR>\n" );
document.write( "CD BD<BR>\n" );
document.write( "CD CD *****\r<BR>\n" );
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document.write( "out of the 36 possible pairings, there are 6 pairings that match.\\r<BR>\n" );
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document.write( "the probability of getting a match is therefore 6 / 36 = 1/6\r<BR>\n" );
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document.write( "it looks like the formula is holding.\r<BR>\n" );
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document.write( "the formula is 1 / C(n,x)\r<BR>\n" );
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document.write( "in this case the formula is 1 / C(4,2) which is equal to 1/6.\r<BR>\n" );
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document.write( "in the case of 3 letter alphabet, the formula was 1 / C(3,2) = 1/3,.\r<BR>\n" );
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document.write( "both cases held true, so we can reasonably assume that the same formula will work with larger numbers.\r<BR>\n" );
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document.write( "the formula of 1 / C(n,x), applied to a 26 letter alphabet, yields 1 / C(26,2) = 1 / 325.\r<BR>\n" );
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document.write( "I believe that's your answer.\r<BR>\n" );
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document.write( "It's hard to prove directly, but it does appear to be good after we tried it on 2 smaller sets of letters.\r<BR>\n" );
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document.write( "we can also try to look at it another way.\r<BR>\n" );
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document.write( "assume you roll 2 dice that have 6 sides each.\r<BR>\n" );
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document.write( "the probability of getting a 1 on the first die is 1/6<BR>\n" );
document.write( "the probability of getting a 1 on the second die is 1/6<BR>\n" );
document.write( "the probability of both die getting a 1 is 1/6 * 1/6 = 1/36.\r<BR>\n" );
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document.write( "the probability of getting a 2  and a 2 is also equal to 1/36<BR>\n" );
document.write( "same for 3,4,5,6.\r<BR>\n" );
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document.write( "the total probability of getting a match on 1 or 2 or 3 or 4 or 5 or 6 is the sum of the individual probabilities is equal to 1/36 * 6 = 6/36 = 1/6.\r<BR>\n" );
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document.write( "now assume the die have 325 sides.\r<BR>\n" );
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document.write( "the same problem becomes 1/325^2 + 1/a325^2 + ....... + 1/325^2 = 325 / 325^2 = 1/325.\r<BR>\n" );
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document.write( "we get the same answer.\r<BR>\n" );
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document.write( "that appears to be your probability based on two sets of reasoning.\r<BR>\n" );
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