document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206041>Question 1206041</A>: <I><SPAN STYLE=\"word-break: break-all;\">   Suppose there is a group of 100 students. 70 of them are in an English class, and 40 of them are in a Math class (so there are 10 students in both). \r<BR>\n" );
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document.write( "Suppose we select a single student at random. Let E= chosen student is in an English class and M=chosen student is in a Math class. <BR>\n" );
document.write( "Compute the following probabilities.<BR>\n" );
document.write( "P(M)=<BR>\n" );
document.write( "P(E)=<BR>\n" );
document.write( "P(M and E)=<BR>\n" );
document.write( "P(M or E)=<BR>\n" );
document.write( "P(M|E)=<BR>\n" );
document.write( "Are these two events independent? \r<BR>\n" );
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document.write( "Now suppose we select two students at random. You can treat this like selecting one, then selecting another, without replacement. \r<BR>\n" );
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document.write( "What is the probability of selecting a student only in an English class first, and then selecting a student only in a Math class second (neither is in both)? \r<BR>\n" );
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document.write( "What is the probability of selecting a student only in a Math class second if my first selection is a student who is only in an English class?<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 #843218 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=843218\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "You posted too many questions. Ideally you should post one problem at a time. <BR>\n" );
document.write( "Multi-part questions are fine as long as they don't get too lengthy.\r<BR>\n" );
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document.write( "I'll answer the first block where it ends with \"P(M|E)\"\r<BR>\n" );
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document.write( "E = set of people taking English<BR>\n" );
document.write( "M = set of people taking Math<BR>\n" );
document.write( "n(E) = number of people taking English<BR>\n" );
document.write( "n(E) = 70<BR>\n" );
document.write( "n(M) = 40<BR>\n" );
document.write( "n(M and E) = 10<BR>\n" );
document.write( "n(M or E) = n(E)+n(M) - n(M and E) ... inclusion-exclusion principle<BR>\n" );
document.write( "n(M or E) = 70+40-10<BR>\n" );
document.write( "n(M or E) = 100\r<BR>\n" );
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document.write( "P(M) = probability a student takes math<BR>\n" );
document.write( "P(M) = n(M)/n(total)<BR>\n" );
document.write( "P(M) = 40/100<BR>\n" );
document.write( "P(M) = <font color=red>2/5</font>\r<BR>\n" );
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document.write( "P(E) = n(E)/n(total)<BR>\n" );
document.write( "P(E) = 70/100<BR>\n" );
document.write( "P(E) = <font color=red>7/10</font>\r<BR>\n" );
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document.write( "P(M and E) = n(M and E)/n(total)<BR>\n" );
document.write( "P(M and E) = 10/100<BR>\n" );
document.write( "P(M and E) = <font color=red>1/10</font>\r<BR>\n" );
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document.write( "P(M or E) = n(M or E)/n(total)<BR>\n" );
document.write( "P(M or E) = 100/100<BR>\n" );
document.write( "P(M or E) = <font color=red>1</font><BR>\n" );
document.write( "A probability of 1 indicates 100% certainty that this event will happen. <BR>\n" );
document.write( "This is because all 100 students either take English, Math, or both.\r<BR>\n" );
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document.write( "P(M | E) = P(M given E)<BR>\n" );
document.write( "P(M | E) = P(M and E)/P(E)<BR>\n" );
document.write( "P(M | E) = P(M and E) ÷ P(E)<BR>\n" );
document.write( "P(M | E) = (1/10) ÷ (7/10)<BR>\n" );
document.write( "P(M | E) = (1/10) * (10/7)<BR>\n" );
document.write( "P(M | E) = 10/70<BR>\n" );
document.write( "P(M | E) = <font color=red>1/7</font><BR>\n" );
document.write( "Another approach:<BR>\n" );
document.write( "P(M | E) = n(M and E)/n(E)<BR>\n" );
document.write( "P(M | E) = 10/70<BR>\n" );
document.write( "P(M | E) = <font color=red>1/7</font><BR>\n" );
document.write( "P(M given E) means we know 100% that the student takes English. <BR>\n" );
document.write( "We focus on the 70 English students. <BR>\n" );
document.write( "Of these 70 students, 10 take math as well. So 10/70 = <font color=red>1/7</font> is the conditional probability we're after.\r<BR>\n" );
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document.write( "Summary:<BR>\n" );
document.write( "P(M) = <font color=red>2/5</font><BR>\n" );
document.write( "P(E) = <font color=red>7/10</font><BR>\n" );
document.write( "P(M and E) = <font color=red>1/10</font><BR>\n" );
document.write( "P(M or E) = <font color=red>1</font><BR>\n" );
document.write( "P(M|E) = <font color=red>1/7</font>\r<BR>\n" );
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document.write( "I'll give a hint about the next question.<BR>\n" );
document.write( "The following equations are true if and only if both events are independent.<ul><li>P(M given E) = P(M)</li><li>P(E given M) = P(E)</li><li>P(M and E) = P(M)*P(E)</li></ul><BR>\n" );
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