document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1048156>Question 1048156</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, my teacher gave us this problem for homework, and I'm having a little bit of trouble solving it:<BR>\n" );
document.write( " Bob has 3 ways to get to work (train, car, and teleport). He independently chooses 1 of the 3 to and from work. <BR>\n" );
document.write( " What is the probability that he teleports at least once during a day?<BR>\n" );
document.write( "I know how to solve it by writing out all the different combinations, but my teacher wants a better, more algebraic-like way, and I just really don't know how to do it. Thanks in advance! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #663763 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</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=663763\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You should be commended for using your own head to solve a problem, especially if you understood the situation in the problem, and knew your answer was right without needing the teacher to tell you so.<BR>\n" );
document.write( "I suspect that your teacher wants you to solve the problem the same way it was shown in class for a similar problem. Sometimes teachers even want to mention certain names and formulas taught in class, maybe using the same symbols in those formulas.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Teleporting at least once during the day, and not teleporting at all during the day are \"mutually exclusive,\" or \"complementary,\"<BR>\n" );
document.write( "so the probability that he teleports at least once during a day,<BR>\n" );
document.write( "plus the probability that he does not teleport at all add to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Can you calculate the probability that he does not teleport at all?<BR>\n" );
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document.write( "The decision to use one or another mode of transportation for the trip to work is supposed to be a \"random event\", so each of the 3 outcomes (train, car, and teleport) is equally probable,<BR>\n" );
document.write( "so the probability of choosing to teleport is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28t%29=1%2F3 ALIGN=MIDDLE  ALT=\"p%28t%29=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The probability that he does not teleport for the trip to work <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28n%29 ALIGN=MIDDLE  ALT=\"p%28n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the complementary event, so the probabilities of teleporting or not teleporting add to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the probability of not teleporting to work is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28n%29=1-1%2F3=2%2F3 ALIGN=MIDDLE  ALT=\"p%28n%29=1-1%2F3=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The same goes for thew second trip of the day (going home),<BR>\n" );
document.write( "so you could say that the probability of not teleporting on the way home is also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28n%29=2%2F3 ALIGN=MIDDLE  ALT=\"p%28n%29=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "The trip to work and the trip home from work are \"independent events,\"<BR>\n" );
document.write( "so the probabilities for the \"compound event\" made of the two trips for a day are the product of the probabilities for each trip.<BR>\n" );
document.write( "So, the probability of not teleporting at all is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29%2A%282%2F3%29=4%2F9 ALIGN=MIDDLE  ALT=\"%282%2F3%29%2A%282%2F3%29=4%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Now it is clear that the probability that he teleports at least once during a day is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-4%2F9=highlight%285%2F9%29 ALIGN=MIDDLE  ALT=\"1-4%2F9=highlight%285%2F9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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