document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209959>Question 1209959</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A club buys tickets for 15 seats in a row for a local production and then sends the tickets randomly to the 15 people who ordered seats. Of these 15 people, 9 are actors and 6 are backstage workers. Find the probability that: a. An actor and backstage worker will receive tickets sitting together. b. The actors will be seated together, and the backstage workers will be seated together. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851245 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=851245\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem into two parts:\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "* We have 15 seats in a row.<BR>\n" );
document.write( "* 9 people are actors (A).<BR>\n" );
document.write( "* 6 people are backstage workers (B).<BR>\n" );
document.write( "* Tickets are assigned randomly.\r<BR>\n" );
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document.write( "**a) Probability an Actor and Backstage Worker Sit Together**\r<BR>\n" );
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document.write( "1. **Calculate the total number of arrangements:**<BR>\n" );
document.write( "   The total number of ways to arrange the 15 people in the 15 seats is 15! (15 factorial).\r<BR>\n" );
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document.write( "2. **Calculate the number of arrangements where an actor and backstage worker sit together:**<BR>\n" );
document.write( "   This is more complex. We'll use the complementary probability: calculate the arrangements where *no* actor and backstage worker sit together, and subtract it from the total.\r<BR>\n" );
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document.write( "   * **Consider pairs:**<BR>\n" );
document.write( "     An actor and backstage worker sit together as a pair (AB or BA).\r<BR>\n" );
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document.write( "   * **Instead, calculate the probability that they are never together:**<BR>\n" );
document.write( "     This calculation is very complex.\r<BR>\n" );
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document.write( "   * **Instead, a simpler approach is to use conditional probability:**<BR>\n" );
document.write( "     * **Probability of the first seat being an actor:** 9/15<BR>\n" );
document.write( "     * **Probability of the second seat being a backstage worker:** 6/14<BR>\n" );
document.write( "     * **Probability of the first seat being a backstage worker:** 6/15<BR>\n" );
document.write( "     * **Probability of the second seat being an actor:** 9/14\r<BR>\n" );
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document.write( "     * **Probability that the first two are an actor and backstage worker:** (9/15)*(6/14) + (6/15)*(9/14) = 54/210 + 54/210 = 108/210\r<BR>\n" );
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document.write( "    * **However, this only works for the first two seats. It is far more complicated to calculate for all seats.**\r<BR>\n" );
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document.write( "    * **Instead, we will use a simpler approach of approximation:**<BR>\n" );
document.write( "     * Consider the number of pairs of seats. There are 14 pairs of adjacent seats.<BR>\n" );
document.write( "     * The probability that any specific pair has an actor and backstage worker is approximately (9/15)(6/14) + (6/15)(9/14) = 108/210 = 18/35.<BR>\n" );
document.write( "     * The probability that *at least one* pair has an actor and backstage worker is difficult to calculate precisely.<BR>\n" );
document.write( "     * However, the probability that a specific pair has an actor and backstage worker is 18/35, so it is likely that they will sit together.\r<BR>\n" );
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document.write( "**b) Probability Actors Sit Together and Backstage Workers Sit Together**\r<BR>\n" );
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document.write( "1. **Treat actors as a single block:**<BR>\n" );
document.write( "   Consider the 9 actors as one block (A) and the 6 backstage workers as another block (B).\r<BR>\n" );
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document.write( "2. **Arrange the blocks:**<BR>\n" );
document.write( "   There are 2! (2 factorial) ways to arrange these two blocks (AB or BA).\r<BR>\n" );
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document.write( "3. **Arrange the actors:**<BR>\n" );
document.write( "   There are 9! ways to arrange the 9 actors within their block.\r<BR>\n" );
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document.write( "4. **Arrange the backstage workers:**<BR>\n" );
document.write( "   There are 6! ways to arrange the 6 backstage workers within their block.\r<BR>\n" );
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document.write( "5. **Calculate the number of favorable arrangements:**<BR>\n" );
document.write( "   The number of favorable arrangements is 2! * 9! * 6!\r<BR>\n" );
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document.write( "6. **Calculate the total number of arrangements:**<BR>\n" );
document.write( "   The total number of arrangements is 15!\r<BR>\n" );
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document.write( "7. **Calculate the probability:**<BR>\n" );
document.write( "   Probability = (2! * 9! * 6!) / 15!\r<BR>\n" );
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document.write( "8. **Simplify:**<BR>\n" );
document.write( "   Probability = (2 * 9! * 6!) / 15! = (2 * 6!) / (15 * 14 * 13 * 12 * 11 * 10) = (2 * 720) / 3603600 = 1440/3603600 = 1/2502.5\r<BR>\n" );
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document.write( "   Probability = 1440/3603600 = 4/10010\r<BR>\n" );
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document.write( "**Answers**\r<BR>\n" );
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document.write( "* **a) Probability an actor and backstage worker sit together:** This is a complex calculation and requires more advanced methods. It is likely a high probability.<BR>\n" );
document.write( "* **b) Probability actors and backstage workers sit together:** 4/10010 or approximately 0.0003996.<BR>\n" );
document.write( " </SPAN>\n" );
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