document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198690>Question 1198690</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are 200 students in a particular graduate program at a state university. Of them, 110 are female and 125 are out- of state students. Of the 110 females, 70 are out  -  of  -  state students. Draw and label a Venn diagram depicting this information. \r<BR>\n" );
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document.write( "Are the events” female” and “ out  -  of  -  state student” independent? Are they mutually exclusive? Explain why or why not.<BR>\n" );
document.write( "If one of these 200 students is selected at random, what is the probability that the student selected is an out  -  of  -  state student given this student is a female?\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 #832325 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=832325\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Here's the Venn Diagram<BR>\n" );
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document.write( "I started with the 70 female out-of-state students in the overlapped region of the two circles.\r<BR>\n" );
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document.write( "Since there are 110 female students, there must be 110-70 = 40 female students who aren't out-of-state. <BR>\n" );
document.write( "In other words, these female students are in-state.<BR>\n" );
document.write( "This value goes inside the circle on the left, but outside the other circle.\r<BR>\n" );
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document.write( "There are 125 out-of-state students, which means 125-70 = 55 out-of-state students aren't female. <BR>\n" );
document.write( "This value goes inside the circle on the right, but outside the other circle.\r<BR>\n" );
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document.write( "The values in either circle add to 40+70+55 = 165 to represent the number of students who are female, out-of-state, or both.<BR>\n" );
document.write( "There are a total of 200 people surveyed, so we have 200-165 = 35 students who are neither female nor out-of-state.<BR>\n" );
document.write( "We'll have this final value outside of both circles.<BR>\n" );
document.write( "As a check, all four values should add to the grand total of 200.<BR>\n" );
document.write( "The rectangle represents the universal set, which means every value is inside it.\r<BR>\n" );
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document.write( "Here is another way to represent those set of values in a two-way table.<BR>\n" );
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document.write( "<tr><TD ></TD><TD >In-State</TD><TD >Out-Of-State</TD></tr><tr><TD >Female</TD><TD >40</TD><TD >70</TD></tr><tr><TD >Male</TD><TD >35</TD><TD >55</TD></tr>\n" );
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document.write( "and this is what it looks like when we have row and column totals<BR>\n" );
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document.write( "<tr><TD ></TD><TD >In-State</TD><TD >Out-Of-State</TD><TD >Total</TD></tr><tr><TD >Female</TD><TD >40</TD><TD >70</TD><TD >110</TD></tr><tr><TD >Male</TD><TD >35</TD><TD >55</TD><TD >90</TD></tr><tr><TD >Total</TD><TD >75</TD><TD >125</TD><TD >200</TD></tr>\n" );
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document.write( "Define these two events<BR>\n" );
document.write( "A = person selected is female<BR>\n" );
document.write( "B = person selected is out-of-state\r<BR>\n" );
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document.write( "From the venn diagram, and also the given info:<BR>\n" );
document.write( "P(A) = 110/200 = 0.55<BR>\n" );
document.write( "P(B) = 125/200 = 0.625<BR>\n" );
document.write( "P(A and B) = 70/200 = 0.35\r<BR>\n" );
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document.write( "If events A and B were independent, then <BR>\n" );
document.write( "P(A and B) = P(A)*P(B)<BR>\n" );
document.write( "would be a true equation.\r<BR>\n" );
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document.write( "Let's see if that's the case<BR>\n" );
document.write( "P(A)*P(B) = 0.55*0.625 = 0.34375<BR>\n" );
document.write( "which is close to the 0.35 calculated earlier, but not quite there.<BR>\n" );
document.write( "Therefore, P(A and B) = P(A)*P(B) is false in this case; meaning A and B <font color=red>are <font size=4><u>not</u></font> independent</font>.\r<BR>\n" );
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document.write( "Now on to see if the events are mutually exclusive.<BR>\n" );
document.write( "A and B are mutually exclusive if and only if<BR>\n" );
document.write( "P(A and B) = 0<BR>\n" );
document.write( "but <BR>\n" );
document.write( "P(A and B) = 0.35 which clearly is nonzero.<BR>\n" );
document.write( "Events A and B are <font color=red>are <font size=4><u>not</u></font> mutually exclusive</font>\r<BR>\n" );
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document.write( "Now onto the last part of this question: <font color=blue>If one of these 200 students is selected at random, what is the probability that the student selected is an out-of-state student given this student is a female?</font>\r<BR>\n" );
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document.write( "The part at the end \"given this student is a female\" tells us that we focus on everyone inside the \"female\" circle of the venn diagram.<BR>\n" );
document.write( "Ignore anyone else.\r<BR>\n" );
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document.write( "Of those 110 women, 70 are out-of-state<BR>\n" );
document.write( "70/110 = 7/11 is the probability of selecting an out-of-state person if we know 100% the person is a woman.\r<BR>\n" );
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document.write( "7/11 = 0.63636 = 63.636% approximately\r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "Summary:\r<BR>\n" );
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document.write( "Question: <font color=blue>Are the events \"female\" and \"out-of-state student\" independent?</font> <BR>\n" );
document.write( "Answer: <font color=red>No</font>, the events are not independent.\r<BR>\n" );
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document.write( "Question: <font color=blue>Are the events \"female\" and \"out-of-state student\" mutually exclusive?</font><BR>\n" );
document.write( "Answer: <font color=red>No</font>, the events are not mutually exclusive.\r<BR>\n" );
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document.write( "Question: <font color=blue>If one of these 200 students is selected at random, what is the probability that the student selected is an out-of-state student given this student is a female?</font><BR>\n" );
document.write( "Answer: <font color=red>7/11</font><BR>\n" );
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