document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171179>Question 1171179</A>: <I><SPAN STYLE=\"word-break: break-all;\">  A survey found that 62% of callers in the United States complain about the service they receive from a call center if they suspect that the agent who handled the call is foreign.11 Given this context, what is the probability that (state your assumptions)\r<BR>\n" );
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document.write( "(a) the next three consecutive callers complain about the service provided by a foreign agent?\r<BR>\n" );
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document.write( "(b) the next two calls produce a complaint, but not the third?\r<BR>\n" );
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document.write( "(c) two out of the next three calls produce a complaint?\r<BR>\n" );
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document.write( "(d) none of the next 10 calls produces a complaint? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850980 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850980\">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( "**Assumptions:**\r<BR>\n" );
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document.write( "1.  **Independence:** We assume that each caller's complaint is independent of the other callers.<BR>\n" );
document.write( "2.  **Constant Probability:** We assume that the probability of a caller complaining about the service provided by a foreign agent is constant at 62% (0.62).\r<BR>\n" );
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document.write( "**Given Information:**\r<BR>\n" );
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document.write( "* Probability of a complaint (p) = 0.62<BR>\n" );
document.write( "* Probability of no complaint (q) = 1 - p = 1 - 0.62 = 0.38\r<BR>\n" );
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document.write( "**Calculations:**\r<BR>\n" );
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document.write( "**(a) The next three consecutive callers complain.**\r<BR>\n" );
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document.write( "* Since the events are independent, we multiply the probabilities:<BR>\n" );
document.write( "    * P(complaint, complaint, complaint) = p * p * p = p^3<BR>\n" );
document.write( "    * P(3 complaints) = (0.62)^3 = 0.238328<BR>\n" );
document.write( "    * Rounded to 4 decimal places: 0.2383\r<BR>\n" );
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document.write( "**(b) The next two calls produce a complaint, but not the third.**\r<BR>\n" );
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document.write( "* P(complaint, complaint, no complaint) = p * p * q = p^2 * q<BR>\n" );
document.write( "    * P(2 complaints, then no complaint) = (0.62)^2 * (0.38) = 0.3844 * 0.38 = 0.146072<BR>\n" );
document.write( "    * Rounded to 4 decimal places: 0.1461\r<BR>\n" );
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document.write( "**(c) Two out of the next three calls produce a complaint.**\r<BR>\n" );
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document.write( "* There are three possible scenarios:<BR>\n" );
document.write( "    * complaint, complaint, no complaint (p * p * q)<BR>\n" );
document.write( "    * complaint, no complaint, complaint (p * q * p)<BR>\n" );
document.write( "    * no complaint, complaint, complaint (q * p * p)<BR>\n" );
document.write( "* Each of these scenarios has the same probability: p^2 * q<BR>\n" );
document.write( "* Since there are three scenarios, we multiply by 3:<BR>\n" );
document.write( "    * P(2 complaints out of 3) = 3 * p^2 * q<BR>\n" );
document.write( "    * P(2 complaints out of 3) = 3 * (0.62)^2 * (0.38) = 3 * 0.146072 = 0.438216<BR>\n" );
document.write( "    * Rounded to 4 decimal places: 0.4382\r<BR>\n" );
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document.write( "**(d) None of the next 10 calls produces a complaint.**\r<BR>\n" );
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document.write( "* P(no complaint) = q = 0.38<BR>\n" );
document.write( "* Since the events are independent, we multiply the probabilities:<BR>\n" );
document.write( "    * P(no complaint for 10 calls) = q^10<BR>\n" );
document.write( "    * P(no complaint for 10 calls) = (0.38)^10 = 0.000062788<BR>\n" );
document.write( "    * Rounded to 4 decimal places: 0.0001<BR>\n" );
document.write( " </SPAN>\n" );
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