document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1180230>Question 1180230</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Q-3: [5+5+5+5 Marks] The classification of faculty at a college is indicated in the following table:\r<BR>\n" );
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document.write( "	               Male   Female	Total\r<BR>\n" );
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document.write( "Professor	        12      3	15\r<BR>\n" );
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document.write( "Associate professor	15	9	24\r<BR>\n" );
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document.write( "Assistant professor	18	8	26\r<BR>\n" );
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document.write( "Instructor       	20	15	35\r<BR>\n" );
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document.write( "Total	                65	35	100\r<BR>\n" );
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document.write( "a)	If  a committee of three faculty members is selected at random what is the probability that:<BR>\n" );
document.write( "i)	All females.<BR>\n" );
document.write( "ii)	A professor and two associate professors.\r<BR>\n" );
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document.write( "b)	 If a faculty member is selected at random from this college, find the probability that:<BR>\n" );
document.write( "i)	The faculty is male, given that the faculty is instructor.<BR>\n" );
document.write( "ii)	The faculty is not professor given that the faculty is 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 #850120 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850120\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Committee of Three Faculty Members:**\r<BR>\n" );
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document.write( "**i) Probability of all females:**\r<BR>\n" );
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document.write( "* Total number of females: 35<BR>\n" );
document.write( "* Total number of faculty: 100<BR>\n" );
document.write( "* Number of ways to choose 3 females: ³⁵C₃ = (35 \* 34 \* 33) / (3 \* 2 \* 1) = 6545<BR>\n" );
document.write( "* Number of ways to choose any 3 faculty: ¹⁰⁰C₃ = (100 \* 99 \* 98) / (3 \* 2 \* 1) = 161700<BR>\n" );
document.write( "* Probability: 6545 / 161700 ≈ 0.0405\r<BR>\n" );
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document.write( "**ii) Probability of a professor and two associate professors:**\r<BR>\n" );
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document.write( "* Number of ways to choose 1 professor: ¹⁵C₁ = 15<BR>\n" );
document.write( "* Number of ways to choose 2 associate professors: ²⁴C₂ = (24 \* 23) / (2 \* 1) = 276<BR>\n" );
document.write( "* Number of ways to choose 1 professor and 2 associate professors: 15 \* 276 = 4140<BR>\n" );
document.write( "* Probability: 4140 / 161700 ≈ 0.0256\r<BR>\n" );
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document.write( "**b) Single Faculty Member Selected:**\r<BR>\n" );
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document.write( "**i) Probability of male, given instructor:**\r<BR>\n" );
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document.write( "* Number of male instructors: 20<BR>\n" );
document.write( "* Total number of instructors: 35<BR>\n" );
document.write( "* Probability: 20 / 35 ≈ 0.5714\r<BR>\n" );
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document.write( "**ii) Probability of not professor, given female:**\r<BR>\n" );
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document.write( "* Total number of female faculty: 35<BR>\n" );
document.write( "* Number of female professors: 3<BR>\n" );
document.write( "* Number of female faculty who are not professors: 35 - 3 = 32<BR>\n" );
document.write( "* Probability: 32 / 35 ≈ 0.9143<BR>\n" );
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