document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1042549>Question 1042549</A>: <I><SPAN STYLE=\"word-break: break-all;\"> One evening, n men enter the restaurant and put their hats at the reception. Each man gets a random hat back <BR>\n" );
document.write( "when going back after having dinner. Find the expected number of men who get their right hat back.\r<BR>\n" );
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document.write( "A. 1\r<BR>\n" );
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document.write( "B. 1/2\r<BR>\n" );
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document.write( "C. 1/n\r<BR>\n" );
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document.write( "D. n/2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #657657 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=657657\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are n hats and each person picks a hat uniformly at random hence each gets their right hat back with probability <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fn ALIGN=MIDDLE  ALT=\"1%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "Since expectation is linear even when the random variables are dependent, it follows that the mean of the total number of persons who <BR>\n" );
document.write( "get their right hat back is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fn+%2B+1%2Fn ALIGN=MIDDLE  ALT=\"1%2Fn+%2B+1%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" >+..+ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fn+=+n%2A%281%2Fn%29+=+highlight%281%29 ALIGN=MIDDLE  ALT=\"1%2Fn+=+n%2A%281%2Fn%29+=+highlight%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "***This is the famous \"hat-matching problem\".  The one given above is the most intuitive but the least rigorous of all the proofs I've seen, <BR>\n" );
document.write( "but does the job well in giving the correct answer.  The other rigorous proofs involve finding the pmf of \"mismatches\", or derangements of n objects (in mathematical parlance) and then applying the definition of expectation of a random variable.***\r<BR>\n" );
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