document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206432>Question 1206432</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An urn contains 5 red and 7 blue balls. Suppose that two balls are selected randomly and with replacement. Let A and B be the events that the first and the second balls are red respectively.<BR>\n" );
document.write( "(a)Is event A and B independent？Show your reasoning.<BR>\n" );
document.write( "(b)If we do the same experiment without replacement, is event A and event B independent? Show your reasoning. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843908 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=843908\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> independence means the probability is not affected by what came before.\r<BR>\n" );
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document.write( "the urn contains 5 red and 7 blue balls.<BR>\n" );
document.write( "you select 2 balls at random, with replacement.\r<BR>\n" );
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document.write( "you want A and B the event of drawing a red ball.\r<BR>\n" );
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document.write( "the first time you draw a ball randomly, the probability that the ball is red is 5 / 12.\r<BR>\n" );
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document.write( "that's because there are 5 red balls in the pot out of a total of 12 balls (5 red plus 7 blue = 12 total).\r<BR>\n" );
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document.write( "after you put the ball back in the pot, the probability of drawing a red ball the second time is also 5 / 12.\r<BR>\n" );
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document.write( "the probability is the same because of the replacement.<BR>\n" );
document.write( "you had 5 red out of 12 the first time and you have 5 red out of 12 the second time.\r<BR>\n" );
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document.write( "that makes the 2 draws independent of each other.\r<BR>\n" );
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document.write( "if you draw the red ball the first time and don't replace it, then the probability of drawing a red ball the second time changes.\r<BR>\n" );
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document.write( "it is not 4 / 11.\r<BR>\n" );
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document.write( "that's because there are now 4 red balls in the pot out of a total of 11.\r<BR>\n" );
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document.write( "the probability of drawing a red ball the second time is different because it was affected by what you draw the second time.\r<BR>\n" );
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document.write( "that makes the probability of drawing a red ball the second time dependent on what happened the first time.\r<BR>\n" );
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document.write( "note that the probability of drawing a red ball the second time is changed whether or not you drew a red ball the first time.\r<BR>\n" );
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document.write( "if you drew a red ball the first time, the probability of drawing a red ball the second time is 4/11.\r<BR>\n" );
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document.write( "if you drew a blue ball the first time, the probability of drawing a red ball the second time is 5/11.\r<BR>\n" );
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document.write( "you still have 5 red balls in the pot, but the total number of balls in the pot changed from 12 to 11.\r<BR>\n" );
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document.write( "bottom line:\r<BR>\n" );
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document.write( "with replacement, probability of drawing a red ball on the second draw is the same as the probability of drawing a red ball on the first draw because it was not affected by what happened on the first draw.\r<BR>\n" );
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document.write( "without replacement, probability of drawing a red ball on the second draw is different from the probability of drawing a red ball on the first draw because it was affected by what happened on the first draw.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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