document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163922>Question 1163922</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let’s assume the following statements are true: Historically, 75% of the five-star football recruits in the nation go to universities in the three most competitive athletic conferences. Historically, five-star recruits get full football scholarships 93% of the time, regardless of which conference they go to. If this pattern holds true for this year’s recruiting class, answer the following:\r<BR>\n" );
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document.write( "a. Based on these numbers, what is the probability that a randomly selected five-star recruit who chooses one of the best three conferences will be offered a full football scholarship? <BR>\n" );
document.write( "b. What are the odds a randomly selected five-star recruit will not select a university from one of the three best conferences? Explain. <BR>\n" );
document.write( "c. Explain whether these are independent or dependent events. Are they Inclusive or exclusive? Explain. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854368 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.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=854368\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Based on the statistics provided, here are the calculations and logical breakdowns for this year's recruiting class:\r<BR>\n" );
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document.write( "### a. Probability of a Scholarship in a Top Conference<BR>\n" );
document.write( "The probability is **93%** (or **0.93**).\r<BR>\n" );
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document.write( "**Reasoning:** The prompt states that five-star recruits get full scholarships 93% of the time **\"regardless of which conference they go to.\"** This implies that the scholarship rate and the conference choice are treated as independent factors in this dataset. Therefore, knowing the recruit chose a top conference does not change the established 93% probability.\r<BR>\n" );
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document.write( "### b. Odds of Not Selecting a Top Conference<BR>\n" );
document.write( "The odds are **1 to 3** (or $1:3$).\r<BR>\n" );
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document.write( "**Explanation:**<BR>\n" );
document.write( "1.  **Probability ($P$):** We know 75% ($3/4$) of recruits choose a top conference. Therefore, the probability of **not** choosing one is 25% ($1/4$).<BR>\n" );
document.write( "2.  **Calculating Odds:** Odds are expressed as the ratio of *Successes* to *Failures* (or in this case, *Not Selecting* vs. *Selecting*).<BR>\n" );
document.write( "    * $\text{Odds} = \frac{P(\text{Not Top})}{P(\text{Top})} = \frac{25\%}{75\%} = \frac{1}{3}$<BR>\n" );
document.write( "3.  **Interpretation:** For every 1 recruit who chooses a school outside the top three conferences, there are 3 recruits who choose a school within them.\r<BR>\n" );
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document.write( "### c. Event Relationships\r<BR>\n" );
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document.write( "**Independent vs. Dependent**<BR>\n" );
document.write( "These are **Independent** events. <BR>\n" );
document.write( "* **Why?** The prompt explicitly states the scholarship rate is 93% \"regardless of which conference they go to.\" This means the outcome of the first event (choosing a conference) has no effect on the probability of the second event (getting a scholarship).\r<BR>\n" );
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document.write( "**Inclusive vs. Exclusive**<BR>\n" );
document.write( "These are **Inclusive** events.<BR>\n" );
document.write( "* **Why?** Mutually exclusive events are things that cannot happen at the same time (like a coin landing on both Heads and Tails). In this scenario, a recruit **can** both choose a top conference **and** receive a full scholarship simultaneously. Since these two outcomes can overlap in a single person, they are inclusive.\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "Do you have a specific recruit's data you're trying to model, or are you looking for the combined probability of both events happening at once? </SPAN>\n" );
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