document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1158866>Question 1158866</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Urn A contains 5 red marbles and 3 white marbles.<BR>\n" );
document.write( "Urn B contains 1 red marble and 2 white marbles.<BR>\n" );
document.write( "A fair die is tossed; if a 3 or 6 appears, a marble is drawn from B and put into A and then a marble is drawn from B.<BR>\n" );
document.write( "a. what is the probability that both marbles are red?<BR>\n" );
document.write( "b. what is the probability that both marbles are white?<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 #854670 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5396)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854670\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I assume that if the number on the top of the die that was tossed is neither 3 nor 6, no marbles are drawn. Your turn is over. better luck next time.<BR>\n" );
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document.write( "Each of the 6 faces/numbers on a fair die has the same probability to appear on top : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F6 ALIGN=MIDDLE  ALT=\"1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The probability of getting a 3 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F6 ALIGN=MIDDLE  ALT=\"1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the probability of getting a 6 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F6 ALIGN=MIDDLE  ALT=\"1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >  for each draw, the probability of getting either a 3 or a 6 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F6%2B1%2F6=2%2F6=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F6%2B1%2F6=2%2F6=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "A teacher might write that as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%283or6%29=1%2F3 ALIGN=MIDDLE  ALT=\"P%283or6%29=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "Let's think of what happens in the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the cases when the game gets to the point when marbles can be drawn.\r<BR>\n" );
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document.write( "Urn B starts with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B2=3 ALIGN=MIDDLE  ALT=\"1%2B2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles.<BR>\n" );
document.write( "After transferring one marble from urn B to urn A,<BR>\n" );
document.write( "urn A will have 9 marbles, either 6red and 3 while or 5red and 4 white,<BR>\n" );
document.write( "depending on the color of the marble drawn from urn B\r<BR>\n" );
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document.write( "The entire play sequence can be diagrammed as follows<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C360%2C0%2C100%2C-60%2C0%2C%0D%0Alocate%281%2C-18.5%2Cstart%29%2Clocate%2821%2C-1%2CDie%29%2C%0D%0Alocate%2839%2C-1%2Curn%29%2Clocate%2845%2C-1%2CB%29%2C%0D%0Alocate%2857%2C-1%2Curn%29%2Clocate%2863%2C-1%2CA%29%2C%0D%0Alocate%2875%2C-1%2CProbability%29%2Clocate%2875%2C-6%2C4%2F6=2%2F3%29%2C%0D%0Aline%289%2C-20%2C20%2C-10%29%2Cline%289%2C-20%2C20%2C-30%29%2C%0D%0Alocate%2820%2C-5%2Cneither%29%2Clocate%2830%2C-5%2C3%29%0D%0Alocate%2821%2C-7%2Cnor%29%2Clocate%2827%2C-7%2C6%29%2C%0D%0Alocate%2823%2C-28%2C3%29%2Clocate%2821%2C-30%2Cor%29%2Clocate%2825%2C-30%2C6%29%2C%0D%0Alocate%2840%2C-21%2Cred%29%2Clocate%2872%2C-14%2C%281%2F3%29%28red%281%2F3%29%29%282%2F3%29=highlight%282%2F27%29%29%2C%0D%0Aline%2828%2C-30%2C38%2C-23%29%2Cline%2828%2C-32%2C38%2C-40%29%2C%0D%0Alocate%2839%2C-39%2Cwhite%29%2Clocate%2872%2C-25%2C%281%2F3%29%28red%281%2F3%29%29%281%2F3%29=1%2F27%29%2C%0D%0Aline%2845%2C-22%2C55%2C-17%29%2Cline%2845%2C-23%2C55%2C-28%29%2C%0D%0Aline%2846%2C-41%2C55%2C-36%29%2Cline%2846%2C-41.5%2C55%2C-46.5%29%2C%0D%0Alocate%2857%2C-15%2Cred%29%2Clocate%2856%2C-26.5%2Cwhite%29%2C%0D%0Alocate%2857%2C-34%2Cred%29%2Clocate%2856%2C-45.5%2Cwhite%29%2C%0D%0Alocate%2814%2C-15%2C%222%2F3%22%29%2Clocate%2814%2C-22%2C%221%2F3%22%29%2C%0D%0Alocate%2833%2C-26%2Cred%28%221%2F3%22%29%29%2Clocate%2833%2C-33%2Cgreen%28%222%2F3%22%29%29%2C%0D%0Alocate%2851%2C-19%2C%222%2F3%22%29%2Clocate%2851%2C-24%2C%221%2F3%22%29%2C%0D%0Alocate%2851%2C-38%2C%225%2F9%22%29%2Clocate%2851%2C-42%2C%224%2F9%22%29%2C%0D%0Alocate%2872%2C-33%2C%281%2F3%29%28green%282%2F3%29%29%285%2F9%29=10%2F81%29%2Clocate%2872%2C-44%2C%281%2F3%29%28green%282%2F3%29%29%284%2F9%29=highlight%288%2F81%29%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "a. what is the probability that both marbles are red?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28red%281%2F3%29%29%282%2F3%29=highlight%282%2F27%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28red%281%2F3%29%29%282%2F3%29=highlight%282%2F27%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b. what is the probability that both marbles are white?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28green%282%2F3%29%29%284%2F9%29=highlight%288%2F81%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28green%282%2F3%29%29%284%2F9%29=highlight%288%2F81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "After the die shows a 3 or a 6, when drawing one of the 3 marbles from urn B, the probability of getting each one of them is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the probability of drawing the only one that is red is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "A teacher could write that as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28%22red%7C3+or+6%22%29=red%281%2F3%29 ALIGN=MIDDLE  ALT=\"P%28%22red%7C3+or+6%22%29=red%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "However, drawing that red marble only happens in  that happens in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%281%2F3%29 ALIGN=MIDDLE  ALT=\"red%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the plays when the die was a 3 or 6.<BR>\n" );
document.write( "It happens in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%2A%28red%281%2F3%29%29=1%2F9 ALIGN=MIDDLE  ALT=\"%281%2F3%29%2A%28red%281%2F3%29%29=1%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the plays.<BR>\n" );
document.write( "A teacher could write that as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28%223+or+6%22%2Cred%29 ALIGN=MIDDLE  ALT=\"P%28%223+or+6%22%2Cred%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%283+or+6%29%2AP%28%22red%7C3+or+6%22%29=%281%2F3%29%2Ared%281%2F3%29=1%2F9 ALIGN=MIDDLE  ALT=\"P%283+or+6%29%2AP%28%22red%7C3+or+6%22%29=%281%2F3%29%2Ared%281%2F3%29=1%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "What happens if the marble drawn from urn B is not red, and what is the probability of that?<BR>\n" );
document.write( "If the marble drawn from urn B is not red, it must be one of the 2 white marbles.<BR>\n" );
document.write( "The probability of that happening after getting a 3 or 6 on the die is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28%22white%7C3+or+6%22%29=green%282%2F3%29 ALIGN=MIDDLE  ALT=\"P%28%22white%7C3+or+6%22%29=green%282%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and that will happen in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282%2F3%29 ALIGN=MIDDLE  ALT=\"green%282%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > of plays when the die showed a 3 or a 6.<BR>\n" );
document.write( "That happens in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%2Agreen%282%2F3%29=2%2F9 ALIGN=MIDDLE  ALT=\"%281%2F3%29%2Agreen%282%2F3%29=2%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the plays.<BR>\n" );
document.write( "A teacher could write that as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28%223+or+6%2Cwhite%22%29=P%28%223+or+6%22%29%2AP%28%22white%7C3+or+6%22%29=%281%2F3%29%2Agreen%282%2F3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F9 ALIGN=MIDDLE  ALT=\"2%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "A teacher would say that red and white are \"mutually exclusive\" events, that it has to be one or the other, so if red happens <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the times , white has to happen the other <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F3=1 ALIGN=MIDDLE  ALT=\"3%2F3=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the total of times.<BR>\n" );
document.write( " <BR>\n" );
document.write( "What happens to urn A, after drawing a marble from urn B?<BR>\n" );
document.write( "The game starts with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2B3=8 ALIGN=MIDDLE  ALT=\"5%2B3=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles in urn A.<BR>\n" );
document.write( "After drawing a marble from urn B and putting it into urn A,<BR>\n" );
document.write( "urn A will contain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2B1=9 ALIGN=MIDDLE  ALT=\"8%2B1=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles,<BR>\n" );
document.write( "and the probability of drawing each of them is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F9 ALIGN=MIDDLE  ALT=\"1%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "We know that if the die shows a 3 or a 6 and the marble drawn from urn B (and added to urn A) is red, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2B1=6%5D ALIGN=MIDDLE  ALT=\"5%2B1=6%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the 9 marbles in urn A will be red and in that case, the probability of drawing one of them would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F9=2%2F3 ALIGN=MIDDLE  ALT=\"6%2F9=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The total probability of getting two red marbles is the product of probabilities of the desired outcome for all 3 steps (the die toss, urn B draw, and urn A draw):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28red%281%2F3%29%29%282%2F3%29=highlight%282%2F27%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28red%281%2F3%29%29%282%2F3%29=highlight%282%2F27%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "If the marble drawn from urn B (and added to urn A) is white, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B1=4 ALIGN=MIDDLE  ALT=\"3%2B1=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the 9 marbles in urn A will be red and the probability of drawing one of them would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F9 ALIGN=MIDDLE  ALT=\"4%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In that case, you got 2 white marbles, and the probability of that is the product of the probabilities for each step<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28green%282%2F3%29%29%284%2F9%29=highlight%288%2F81%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28green%282%2F3%29%29%284%2F9%29=highlight%288%2F81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );