document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1137820>Question 1137820</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that a single fair die is rolled one time.  Let events A and B be defined as:\r<BR>\n" );
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document.write( "A  =  { 1, 4, 5 }      and    B  =  { 2, 4, 5, 6 }\r<BR>\n" );
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document.write( "Are A and B independent events?  Show why or why not?  I cant figure this problem out<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 #755706 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=755706\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "I'm assuming the set notation refers to the outcome of rolling the die<BR>\n" );
document.write( "A = {1,4,5} = rolling a 1, 4 or 5<BR>\n" );
document.write( "B = {2,4,5,6} = rolling a 2, 4, 5, or 6\r<BR>\n" );
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document.write( "There are 3 ways to roll an outcome in event A, out of 6 total, so<BR>\n" );
document.write( "P(A) = 3/6 = 1/2\r<BR>\n" );
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document.write( "There are 4 items in set B, so<BR>\n" );
document.write( "P(B) = 4/6 = 2/3\r<BR>\n" );
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document.write( "If A and B were independent, then this equation would be true<BR>\n" );
document.write( "P(A and B) = P(A)*P(B)<BR>\n" );
document.write( "P(A and B) = (1/2)*(2/3)<BR>\n" );
document.write( "P(A and B) = 1/3<BR>\n" );
document.write( "We'll keep this value in mind\r<BR>\n" );
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document.write( "Let <BR>\n" );
document.write( "C = the event in which both A and B happen at the same time<BR>\n" );
document.write( "In other words, event C is rolling either a 4 or 5 since these values are found in both set A = {1,<font color=blue>4,5</font>} and set B = {2,<font color=blue>4,5</font>,6}. We can say that set C is the intersection of A and B<BR>\n" );
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document.write( "C = {4,5}<BR>\n" );
document.write( "We have 2 ways to get a value in set C, out of 6 ways to roll a die<BR>\n" );
document.write( "P(C) = 2/6<BR>\n" );
document.write( "P(C) = 1/3<BR>\n" );
document.write( "we get 1/3 just like with the previous computation\r<BR>\n" );
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document.write( "We see that P(A and B) = P(C) = P(A)*P(B) is true, so therefore we have shown A and B are independent events.<BR>\n" );
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