document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1097303>Question 1097303</A>: <I><SPAN STYLE=\"word-break: break-all;\"> SHOW THAT IF B IS A SUBSET OF A, THEN P(B)<= P(A) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #711952 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=711952\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "U = universal set\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's say that the universal set U has p items inside it where p is some positive integer\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U = \{a_{1},a_{2},\ldots, a_{p}\}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Define positive integers m and n such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+%3C=+n+%3C=+p ALIGN=MIDDLE  ALT=\"m+%3C=+n+%3C=+p\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Set A has n items inside it\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A = \{a_{1},a_{2},\ldots, a_{n}\}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "while set B has m items inside it\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large B = \{a_{1},a_{2},\ldots, a_{m}\}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Every item inside set B is also found in set A. This makes set B a subset of set A.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The probability of selecting an item from set A is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2Fp ALIGN=MIDDLE  ALT=\"n%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" > since we have n items from set A and p items from set U.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The probability of selecting an item from set B is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Fp ALIGN=MIDDLE  ALT=\"m%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" > since we have m items from set B and p items from set U.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+%3C=+n ALIGN=MIDDLE  ALT=\"m+%3C=+n\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Fp+%3C=+n%2Fp ALIGN=MIDDLE  ALT=\"m%2Fp+%3C=+n%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "We can divide both sides of the first inequality (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+%3C=+n ALIGN=MIDDLE  ALT=\"m+%3C=+n\"   DISABLED_style= \"max-width:90%; height:auto;\" >) by p to get the second inquality (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Fp+%3C=+n%2Fp ALIGN=MIDDLE  ALT=\"m%2Fp+%3C=+n%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" >). <BR>\n" );
document.write( "The inequality sign will not flip because p is a positive number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So this proves that P(B) <= P(A) is true, since,<BR>\n" );
document.write( "P(A) = n/p<BR>\n" );
document.write( "P(B) = m/p\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );