document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201611>Question 1201611</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What NCAA college basketball conferences have the higher probability of having a <BR>\n" );
document.write( "team play in college basketball’s national championship game? Over the last 20 years, <BR>\n" );
document.write( "the Atlantic Coast Conference (ACC) ranks first by having a team in the championship <BR>\n" );
document.write( "game 10 times. The South-eastern Conference (SEC) ranks second by having a team in <BR>\n" );
document.write( "the championship game 8 times. However, these two conferences have both had teams <BR>\n" );
document.write( "in the championship game only one time, when Arkansas (SEC) beat Duke (ACC) 76  -  70 in 1994 (NCAA website, April 2009). Use these data to estimate the following <BR>\n" );
document.write( "probabilities.<BR>\n" );
document.write( "a) What is the probability the ACC will have a team in the championship game?<BR>\n" );
document.write( "b) What is the probability the SEC will have team in the championship game?<BR>\n" );
document.write( "c) What is the probability the ACC and SEC will both have teams in the championship<BR>\n" );
document.write( "game?<BR>\n" );
document.write( "d) What is the probability at least one team from these two conferences will be in the<BR>\n" );
document.write( "championship game? That is, what is the probability a team from the ACC or SEC <BR>\n" );
document.write( "will play in the championship game?<BR>\n" );
document.write( "e) What is the probability that the championship game will not a have team from one <BR>\n" );
document.write( "of these two conferences? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848022 by </B><A HREF=/tutors/aboutme.mpl?userid=asinus><B>asinus(45)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=asinus><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=848022\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To estimate the probabilities based on the data provided, we will use the following information:\r<BR>\n" );
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document.write( "- **Atlantic Coast Conference (ACC)**: 10 appearances in the championship game over 20 years.<BR>\n" );
document.write( "- **Southeastern Conference (SEC)**: 8 appearances in the championship game over 20 years.<BR>\n" );
document.write( "- **Both conferences had a team in the championship game together only once** (1994).\r<BR>\n" );
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document.write( "### Step 1: Calculate the Probabilities\r<BR>\n" );
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document.write( "**Total Years**: 20\r<BR>\n" );
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document.write( "#### a) Probability the ACC will have a team in the championship game\r<BR>\n" );
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document.write( "The probability $ P(\text{ACC}) $ is calculated as:\r<BR>\n" );
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document.write( "$$<BR>\n" );
document.write( "P(\text{ACC}) = \frac{\text{Number of times ACC had a team}}{\text{Total years}} = \frac{10}{20} = 0.5<BR>\n" );
document.write( "$$\r<BR>\n" );
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document.write( "#### b) Probability the SEC will have a team in the championship game\r<BR>\n" );
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document.write( "The probability $ P(\text{SEC}) $ is calculated as:\r<BR>\n" );
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document.write( "$$<BR>\n" );
document.write( "P(\text{SEC}) = \frac{\text{Number of times SEC had a team}}{\text{Total years}} = \frac{8}{20} = 0.4<BR>\n" );
document.write( "$$\r<BR>\n" );
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document.write( "#### c) Probability the ACC and SEC will both have teams in the championship game\r<BR>\n" );
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document.write( "Since both conferences had a team in the championship game together only once, the probability $ P(\text{ACC} \cap \text{SEC}) $ is:\r<BR>\n" );
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document.write( "$$<BR>\n" );
document.write( "P(\text{ACC} \cap \text{SEC}) = \frac{1}{20} = 0.05<BR>\n" );
document.write( "$$\r<BR>\n" );
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document.write( "#### d) Probability at least one team from these two conferences will be in the championship game\r<BR>\n" );
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document.write( "To find the probability that at least one team from the ACC or SEC will be in the championship game, we can use the formula for the union of two events:\r<BR>\n" );
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document.write( "$$<BR>\n" );
document.write( "P(\text{ACC} \cup \text{SEC}) = P(\text{ACC}) + P(\text{SEC}) - P(\text{ACC} \cap \text{SEC})<BR>\n" );
document.write( "$$\r<BR>\n" );
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document.write( "Substituting the values we calculated:\r<BR>\n" );
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document.write( "$$<BR>\n" );
document.write( "P(\text{ACC} \cup \text{SEC}) = 0.5 + 0.4 - 0.05 = 0.85<BR>\n" );
document.write( "$$\r<BR>\n" );
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document.write( "#### e) Probability that the championship game will not have a team from one of these two conferences\r<BR>\n" );
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document.write( "The probability that neither the ACC nor the SEC has a team in the championship game is the complement of the probability that at least one of them does:\r<BR>\n" );
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document.write( "$$<BR>\n" );
document.write( "P(\text{Neither ACC nor SEC}) = 1 - P(\text{ACC} \cup \text{SEC}) = 1 - 0.85 = 0.15<BR>\n" );
document.write( "$$\r<BR>\n" );
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document.write( "### Summary of Probabilities\r<BR>\n" );
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document.write( "- a) $ P(\text{ACC}) = 0.5 $<BR>\n" );
document.write( "- b) $ P(\text{SEC}) = 0.4 $<BR>\n" );
document.write( "- c) $ P(\text{ACC} \cap \text{SEC}) = 0.05 $<BR>\n" );
document.write( "- d) $ P(\text{ACC} \cup \text{SEC}) = 0.85 $<BR>\n" );
document.write( "- e) $ P(\text{Neither ACC nor SEC}) = 0.15 $ \r<BR>\n" );
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document.write( "These probabilities provide a clear picture of the likelihood of teams from the ACC and SEC participating in the NCAA basketball championship game based on historical data. </SPAN>\n" );
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