document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176351>Question 1176351</A>: <I><SPAN STYLE=\"word-break: break-all;\"> During a 52-week period, a company paid overtime wages for 18 weeks and hired temporary help for 9 weeks. During 5 weeks, the company paid overtime and hired temporary help. Let: O = overtime, T = temporary help.\r<BR>\n" );
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document.write( "a. What is the probability P(O OR T) ? <BR>\n" );
document.write( "  Round your answer to the nearest hundredth.\r<BR>\n" );
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document.write( "b.  Are O and T independent? \r<BR>\n" );
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document.write( "c. Are O and T mutually exclusive?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850581 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=850581\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**Given Information:**\r<BR>\n" );
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document.write( "* Total weeks: 52<BR>\n" );
document.write( "* Overtime (O) weeks: 18<BR>\n" );
document.write( "* Temporary help (T) weeks: 9<BR>\n" );
document.write( "* Overtime and temporary help (O ∩ T) weeks: 5\r<BR>\n" );
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document.write( "**a) What is the probability P(O OR T)?**\r<BR>\n" );
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document.write( "We'll use the formula for the probability of the union of two events:\r<BR>\n" );
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document.write( "* P(O ∪ T) = P(O) + P(T) - P(O ∩ T)\r<BR>\n" );
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document.write( "1.  **Calculate P(O):**<BR>\n" );
document.write( "    * P(O) = Number of overtime weeks / Total weeks<BR>\n" );
document.write( "    * P(O) = 18 / 52\r<BR>\n" );
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document.write( "2.  **Calculate P(T):**<BR>\n" );
document.write( "    * P(T) = Number of temporary help weeks / Total weeks<BR>\n" );
document.write( "    * P(T) = 9 / 52\r<BR>\n" );
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document.write( "3.  **Calculate P(O ∩ T):**<BR>\n" );
document.write( "    * P(O ∩ T) = Number of weeks with both / Total weeks<BR>\n" );
document.write( "    * P(O ∩ T) = 5 / 52\r<BR>\n" );
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document.write( "4.  **Calculate P(O ∪ T):**<BR>\n" );
document.write( "    * P(O ∪ T) = (18 / 52) + (9 / 52) - (5 / 52)<BR>\n" );
document.write( "    * P(O ∪ T) = (18 + 9 - 5) / 52<BR>\n" );
document.write( "    * P(O ∪ T) = 22 / 52<BR>\n" );
document.write( "    * P(O ∪ T) = 11 / 26<BR>\n" );
document.write( "    * P(O ∪ T) ≈ 0.4230769\r<BR>\n" );
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document.write( "5.  **Round to the nearest hundredth:**<BR>\n" );
document.write( "    * P(O ∪ T) ≈ 0.42\r<BR>\n" );
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document.write( "**b) Are O and T independent?**\r<BR>\n" );
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document.write( "Two events are independent if P(O ∩ T) = P(O) * P(T).\r<BR>\n" );
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document.write( "1.  **Calculate P(O) * P(T):**<BR>\n" );
document.write( "    * P(O) * P(T) = (18 / 52) * (9 / 52)<BR>\n" );
document.write( "    * P(O) * P(T) = 162 / 2704<BR>\n" );
document.write( "    * P(O) * P(T) ≈ 0.0599\r<BR>\n" );
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document.write( "2.  **Compare P(O ∩ T) and P(O) * P(T):**<BR>\n" );
document.write( "    * P(O ∩ T) = 5 / 52 ≈ 0.0961<BR>\n" );
document.write( "    * P(O) * P(T) ≈ 0.0599\r<BR>\n" );
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document.write( "3.  **Conclusion:**<BR>\n" );
document.write( "    * Since P(O ∩ T) ≠ P(O) * P(T), the events O and T are **not independent**.\r<BR>\n" );
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document.write( "**c) Are O and T mutually exclusive?**\r<BR>\n" );
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document.write( "Two events are mutually exclusive if P(O ∩ T) = 0.\r<BR>\n" );
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document.write( "1.  **Check P(O ∩ T):**<BR>\n" );
document.write( "    * P(O ∩ T) = 5 / 52\r<BR>\n" );
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document.write( "2.  **Conclusion:**<BR>\n" );
document.write( "    * Since P(O ∩ T) ≠ 0, the events O and T are **not mutually exclusive**.\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "* **a) P(O OR T) ≈ 0.42**<BR>\n" );
document.write( "* **b) No, O and T are not independent.**<BR>\n" );
document.write( "* **c) No, O and T are not mutually exclusive.**<BR>\n" );
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