document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208266>Question 1208266</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Good day. please help me solve this problems.<BR>\n" );
document.write( "Consider a bag containing 5 blue marbles and 3 green marbles.\r<BR>\n" );
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document.write( "I.	Construct a tree diagram to represent all possible outcomes of drawing two marbles when the marble is replaced after each draw.			[4 marks]\r<BR>\n" );
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document.write( "II.	What is the probability of drawing two blue marbles in a row?		[2 marks]\r<BR>\n" );
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document.write( "III.	What is the probability of selecting a colour of each?			[2 marks]<BR>\n" );
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document.write( "IV.	Construct a tree diagram to represent all possible outcomes of drawing two marbles without replacement.						[4 marks]<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 #846578 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=846578\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part I\r<BR>\n" );
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document.write( "This is one way to draw the probability tree diagram.<BR>\n" );
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document.write( "This style has it flow from left to right.<BR>\n" );
document.write( "Another diagram style would have it flow from top to bottom (rotate the diagram shown above 90 degrees clockwise).\r<BR>\n" );
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document.write( "The probability tree should be self-explanatory.<BR>\n" );
document.write( "Perhaps the only explanation that's needed is each item in the \"outcome\" column is the result of multiplying the fractions along the particular pathway.\r<BR>\n" );
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document.write( "For instance, multiply the fractions along the upper most path to get (5/8)*(5/8) = 25/64 which is the probability of selecting 2 blue marbles in a row where replacement is done.\r<BR>\n" );
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document.write( "The list of possible outcomes are<BR>\n" );
document.write( "P(BB) = 25/64<BR>\n" );
document.write( "P(BG) = 15/64<BR>\n" );
document.write( "P(GB) = 15/64<BR>\n" );
document.write( "P(GG) = 9/64\r<BR>\n" );
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document.write( "Optionally we can make a table of outcomes<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >2nd is B</TD><TD >2nd is G</TD></tr><tr><TD >1st is B</TD><TD >25/64</TD><TD >15/64</TD></tr><tr><TD >1st is G</TD><TD >15/64</TD><TD >9/64</TD></tr>\n" );
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document.write( "Note that P(BB)+P(BG)+P(GB)+P(GG) = (25/64)+(15/64)+(15/64)+(9/64) = 1\r<BR>\n" );
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document.write( "More Practice<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.mathsisfun.com/data/probability-tree-diagrams.html\">https://www.mathsisfun.com/data/probability-tree-diagrams.html</a>\r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Part II\r<BR>\n" );
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document.write( "I'll assume replacement is applied.<BR>\n" );
document.write( "P(BB) = probability of two blue marbles in a row with replacement<BR>\n" );
document.write( "P(BB) = <font color=red>25/64</font> was calculated back in part I. \r<BR>\n" );
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document.write( "Highlight the upper most path for the two blue marbles.<BR>\n" );
document.write( "Multiply the fractions along this path.\r<BR>\n" );
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document.write( "Part III\r<BR>\n" );
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document.write( "I'll assume replacement is applied.\r<BR>\n" );
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document.write( "There are two methods we could use.<BR>\n" );
document.write( "The first method is most direct and quickest.<BR>\n" );
document.write( "P(one of each color) = P(BG) + P(GB)<BR>\n" );
document.write( "P(one of each color) = (15/64)+(15/64)<BR>\n" );
document.write( "P(one of each color) = (15+15)/64<BR>\n" );
document.write( "P(one of each color) = 30/64<BR>\n" );
document.write( "P(one of each color) = (2*15)/(2*32)<BR>\n" );
document.write( "P(one of each color) = <font color=red>15/32</font>\r<BR>\n" );
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document.write( "Here is another approach.<BR>\n" );
document.write( "P(two of same color) = P(2 blue OR 2 green)<BR>\n" );
document.write( "P(two of same color) = P(BB)+P(GG) <BR>\n" );
document.write( "P(two of same color) = (25/64)+(9/64)<BR>\n" );
document.write( "P(two of same color) = (25+9)/64 <BR>\n" );
document.write( "P(two of same color) = 34/64<BR>\n" );
document.write( "P(two of same color) = (2*17)/(2*32)<BR>\n" );
document.write( "P(two of same color) = 17/32<BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "P(one of each color) = 1 - P(two of same color)<BR>\n" );
document.write( "P(one of each color) = 1 - (17/32)<BR>\n" );
document.write( "P(one of each color) = (32/32) - (17/32)<BR>\n" );
document.write( "P(one of each color) = (32-17)/32<BR>\n" );
document.write( "P(one of each color) = <font color=red>15/32</font>\r<BR>\n" );
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document.write( "Part IV\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-2.5%2C5.2%2C-3.5%2C5.5%2C%0D%0A%0D%0Alocate%280.5%2C5%2Cmatrix%281%2C2%2C%22Without%22%2C%22Replacement%22%29%29%2C%0D%0Alocate%28-1%2B0.5%2C4%2C%22first%22%29%2C%0D%0Alocate%281%2B0.5%2C4%2C%22second%22%29%2C%0D%0A%0D%0Aline%282%2C-3%2C0%2C-2%29%2Cline%282%2C-1%2C0%2C-2%29%2Cline%282%2C1%2C0%2C2%29%2Cline%282%2C3%2C0%2C2%29%2Cline%280%2C-2%2C-2%2C0%29%2Cline%280%2C2%2C-2%2C0%29%2C%0D%0A%0D%0Alocate%282.28%2C-2.64-0.2%2C%22G%22%29%2Clocate%282.22%2C-0.54-0.2%2C%22B%22%29%2Clocate%282.3%2C1.34-0.2%2C%22G%22%29%2Clocate%282.24%2C3.34-0.2%2C%22B%22%29%2Clocate%280%2C-1.62%2B0.1%2C%22G%22%29%2Clocate%280%2C2.38%2B0.1%2C%22B%22%29%2C%0D%0A%0D%0Alocate%281%2C-2.18%2C%222%2F7%22%29%2Clocate%280.78%2C-0.98%2B0.1%2C%225%2F7%22%29%2Clocate%281.14%2C1.8%2C%223%2F7%22%29%2Clocate%281%2C3%2B0.1%2C%224%2F7%22%29%2Clocate%28-0.94%2C-0.72%2C%223%2F8%22%29%2Clocate%28-1.12%2C1.28%2B0.5%2C%225%2F8%22%29%2C%0D%0A%0D%0Alocate%283.5%2C4%2C%22outcome%22%29%2C%0D%0Alocate%283.5%2C3.15%2C%22P%28BB%29=5%2F14%22%29%2C%0D%0Alocate%283.5%2C1.15%2C%22P%28BG%29=15%2F56%22%29%2C%0D%0Alocate%283.5%2C-0.85%2C%22P%28GB%29=15%2F56%22%29%2C%0D%0Alocate%283.5%2C-2.85%2C%22P%28GG%29=3%2F28%22%29%2C%0D%0A%0D%0Alocate%28-2.2%2C-2.5%2Cmatrix%281%2C3%2C%22B%22%2C%22=%22%2C%22blue%22%29%29%2C%0D%0Alocate%28-2.2%2C-3%2Cmatrix%281%2C3%2C%22G%22%2C%22=%22%2C%22green%22%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This probability tree is very similar to the tree in part I.<BR>\n" );
document.write( "The 5/8 and 3/8 in the 1st column, or left-most column, are the same as before.\r<BR>\n" );
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document.write( "The fractions in the 2nd column are new. The 7s in the denominator refer to the number of marbles remaining after the 1st selection is not replaced.<BR>\n" );
document.write( "Subtract 1 from the numerator only when you are selecting two of the same color.\r<BR>\n" );
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document.write( "To get each item in the \"outcome\" column, you follow the same idea as mentioned earlier. <BR>\n" );
document.write( "Example: Multiplying along the top most pathway yields (5/8)*(4/7) = 20/56 = 5/14 which is the probability of getting two blue marbles in a row when replacement is not applied.<BR>\n" );
document.write( "Use this idea for the other three pathways.\r<BR>\n" );
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document.write( "Here is an optional table of outcomes<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >2nd is B</TD><TD >2nd is G</TD></tr><tr><TD >1st is B</TD><TD >5/14</TD><TD >15/56</TD></tr><tr><TD >1st is G</TD><TD >15/56</TD><TD >3/28</TD></tr>\n" );
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document.write( "Or we can list them out like so<BR>\n" );
document.write( "P(BB) = 5/14<BR>\n" );
document.write( "P(BG) = 15/56<BR>\n" );
document.write( "P(GB) = 15/56<BR>\n" );
document.write( "P(GG) = 3/28\r<BR>\n" );
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document.write( "Note that P(BB)+P(BG)+P(GB)+P(GG) = (5/14)+(15/56)+(15/56)+(3/28) = 1\r<BR>\n" );
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document.write( "Summary of answers:<ol type=\"I\"><li>Diagram is shown above.</li><li>probability of two blue in a row = <font color=red>25/64</font> (with replacement)</li><li>probability one of each color = <font color=red>15/32</font> (with replacement)</li><li>Diagram is shown above.</li></ol><BR>\n" );
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