document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191712>Question 1191712</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The probability of rain on first 3 days of august is as follows:<BR>\n" );
document.write( "Day                        probability gain<BR>\n" );
document.write( "first day                              3/10<BR>\n" );
document.write( "2nd day                         if rain on day 1 is 5/10<BR>\n" );
document.write( "                            if no rain of day 1 is 2/10<BR>\n" );
document.write( "3rd day                      if rain on day 1 and day 2 are 9/10<BR>\n" );
document.write( "                            if rain on day 1 only is 5/10<BR>\n" );
document.write( "                             if rain on day 2 only is 3/10<BR>\n" );
document.write( "                             if no rain on day 1 and day 2 are 5/10\r<BR>\n" );
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document.write( "what is the probability of rain on atleast 2 out of 3 days?\r<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 #823516 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=823516\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The given probabilities, and the probabilities that can be deduced from the given probabilities, are the following:<br><BR>\n" );
document.write( "P(rain on day 1) = 0.3 (given)<BR>\n" );
document.write( "P(no rain on day 1) = 1-0.3 = 0.7 (deduced)<br><BR>\n" );
document.write( "P(rain on day 2 given rain on day 1) = 0.5 (given)<BR>\n" );
document.write( "P(no rain on day 2 given rain on day 1) = 1-0.5 = 0.5 (deduced)<br><BR>\n" );
document.write( "P(rain on day 2 given no rain on day 1) = 0.2 (given)<BR>\n" );
document.write( "P(no rain on day 2 given no rain on day 1) = 1-0.2 = 0.8 (deduced)<br><BR>\n" );
document.write( "P(rain on day 3 given rain on days 1 and 2) = 0.9 (given)<BR>\n" );
document.write( "P(no rain on day 3 given rain on days 1 and 2) = 1-0.9 = 0.1 (deduced)<br><BR>\n" );
document.write( "P(rain on day 3 given rain only on day 1) = 0.5 (given)<BR>\n" );
document.write( "P(no rain on day 3 given rain only on day 1) = 1-0.5 = 0.5 (deduced)<br><BR>\n" );
document.write( "P(rain on day 3 given rain only on day 2) = 0.3 (given)<BR>\n" );
document.write( "P(no rain on day 3 given rain only on day 2) = 1-0.3 = 0.7 (deduced)<br><BR>\n" );
document.write( "P(rain on day 3 given no rain on days 1 or 2) = 0.5 (given)<BR>\n" );
document.write( "P(no rain on day 3 given no rain on days 1 or 2) = 1-0.5 = 0.5 (deduced)<br><BR>\n" );
document.write( "We could use those probabilities to answer the question directly,  However, it is useful to calculate the probabilities of all possible outcomes to verify that their sum is 1; that will give us confidence we have done the calculations correctly.<br><BR>\n" );
document.write( "Using \"Y\" (yes) or \"N\" (no) to indicate whether or not there is rain on each of three days, we have the following probabilities:<br><BR>\n" );
document.write( "P(YYY) = (.3)(.5)(.9) = .135<BR>\n" );
document.write( "P(YYN) = (.3)(.5)(.1) = .015<BR>\n" );
document.write( "P(YNY) = (.3)(.5)(.5) = .075<BR>\n" );
document.write( "P(YNN) = (.3)(.5)(.5) = .075<BR>\n" );
document.write( "P(NYY) = (.7)(.2)(.3) = .042<BR>\n" );
document.write( "P(NYN) = (.7)(.2)(.7) = .098<BR>\n" );
document.write( "P(NNY) = (.7)(.8)(.5) = .280<BR>\n" );
document.write( "P(NNN) = (.7)(.8)(.5) = .280<br><BR>\n" );
document.write( "The sum of those probabilities is 1, so it is very likely we have done the calculations correctly.<br><BR>\n" );
document.write( "Then the answer to the problem is the sum of the probabilities that it rains on either 2 or all 3 of the 3 days:<br><BR>\n" );
document.write( "ANSWER: P(either 2 or 3 days of rain) = .135+.015+.075+.042 = .267<br>\r<BR>\n" );
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