document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196440>Question 1196440</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Draw an appropriate tree diagram, and use the multiplication principle to calculate the probabilities of all the outcomes. HINT [See Example 3.]<BR>\n" );
document.write( "Your auto rental company rents out 30 small cars, 22 luxury sedans, and 48 slightly damaged \"budget\" vehicles. The small cars break down 12% of the time, the luxury sedans break down 9% of the time, and the \"budget\" cars break down 40% of the time.<BR>\n" );
document.write( "P(Small and breaks down)	=	<BR>\n" );
document.write( "P(Small and does not break down)	=	<BR>\n" );
document.write( "P(Luxury and breaks down)	=	<BR>\n" );
document.write( "P(Luxury and does not break down)	=	<BR>\n" );
document.write( "P(Budget and breaks down)	=	<BR>\n" );
document.write( "P(Budget and does not break down)	=	<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 #829269 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=829269\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Event S = customer chose a small car<BR>\n" );
document.write( "Event L = customer chose a luxury car<BR>\n" );
document.write( "Event B = customer chose a budget car<BR>\n" );
document.write( "Event R = car breaks down<BR>\n" );
document.write( "Event ~R = car does not break down\r<BR>\n" );
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document.write( "I'll draw a partial tree diagram, and let you fill out the rest.<BR>\n" );
document.write( "See below.\r<BR>\n" );
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document.write( "The top row of branches represents S, L and B to represent \"small\", \"luxury\", and \"budget\" in that exact order.<BR>\n" );
document.write( "The decimal numbers next to each branch represents the probability of randomly selecting that type of car\r<BR>\n" );
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document.write( "30 small + 22 luxury + 48 budget = 100 cars total<BR>\n" );
document.write( "30/100 = 0.30 is the probability of selecting a small car<BR>\n" );
document.write( "22/100 = 0.22 is the probability of selecting a luxury car<BR>\n" );
document.write( "48/100 = 0.48 is the probability of selecting a budget car\r<BR>\n" );
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document.write( "Then for each of those 3 branches, we'll have 2 branches. That means we have 3*2 = 6 branches total. <BR>\n" );
document.write( "The 2 additional branches represent \"breaks down\" and \"does not break down\", which I'll shorten to R and ~R respectively.\r<BR>\n" );
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document.write( "Keep in mind the probability P(R) changes depending on which car you selected. <BR>\n" );
document.write( "For more information, check out the concept of \"conditional probability\".<BR>\n" );
document.write( "P(R) = 0.12 if event S happens<BR>\n" );
document.write( "P(R) = 0.09 if event L happens<BR>\n" );
document.write( "P(R) = 0.40 if event B happens<BR>\n" );
document.write( "their associated complementary probabilities are...<BR>\n" );
document.write( "P(~R) = 1-0.12 = 0.88 if event S happens<BR>\n" );
document.write( "P(~R) = 1-0.09 = 0.91 if event L happens<BR>\n" );
document.write( "P(~R) = 1-0.40 = 0.60 if event B happens\r<BR>\n" );
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document.write( "Rephrased in slightly different notation<BR>\n" );
document.write( "P(R given S) = 0.12<BR>\n" );
document.write( "P(R given L) = 0.09<BR>\n" );
document.write( "P(R given B) = 0.40<BR>\n" );
document.write( "P(~R given S) = 0.88<BR>\n" );
document.write( "P(~R given L) = 0.91<BR>\n" );
document.write( "P(~R given B) = 0.60\r<BR>\n" );
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document.write( "Those 6 new decimal values will label each of the 6 new branches in the second row. \r<BR>\n" );
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document.write( "Here's the tree diagram so far<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src = \"https://i.imgur.com/HBaVvtm.png\">\r<BR>\n" );
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document.write( "Next, we'll multiply values along the branches to find the answers we're after.<BR>\n" );
document.write( "Along path S and path R directly underneath it, we multiply the values to get 0.30*0.12 = 0.036; the result is placed at the very bottom of this pathway\r<BR>\n" );
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document.write( "Another example: 0.30*0.88 = 0.264 when following the path from S to ~R\r<BR>\n" );
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document.write( "Here's the slightly more updated version of the tree diagram<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src = \"https://i.imgur.com/IFb0lf8.png\">\r<BR>\n" );
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document.write( "I'll let you take over from here. Your task is to fill out the remaining empty boxes following similar steps shown above.<BR>\n" );
document.write( "Once the tree diagram is completed, you'll have the answers needed.\r<BR>\n" );
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document.write( "For instance, the answer to P(small and breaks down) is the value 0.036 found earlier. This is because we followed the S pathway and R pathway.<BR>\n" );
document.write( "There's a 3.6% chance of selecting a small car and having it break down. \r<BR>\n" );
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document.write( "The answer to P(small and does not break down) would be 0.264 since it follows the S to ~R pathway. \r<BR>\n" );
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document.write( "Extra info:\r<BR>\n" );
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document.write( "Notice how<BR>\n" );
document.write( "P(S and R) + P(S and ~R) = P(S)<BR>\n" );
document.write( "0.036 + 0.264 = 0.30<BR>\n" );
document.write( "This is not a coincidence.<BR>\n" );
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