document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193223>Question 1193223</A>: <I><SPAN STYLE=\"word-break: break-all;\"> At a restaurant, the time a customer has to wait before being seated follows an exponential distribution and is given by the probability density function (pdf):\r<BR>\n" );
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document.write( "f(t)=1e−1t<BR>\n" );
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document.write( "Find the probability that a customer will have to wait at least 4 minutes for a table. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848850 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=848850\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Identify the Distribution**\r<BR>\n" );
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document.write( "* The given probability density function (PDF) <BR>\n" );
document.write( "   f(t) = 1/e * e^(-t) <BR>\n" );
document.write( "   represents an **exponential distribution** with a rate parameter (λ) of 1.\r<BR>\n" );
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document.write( "**2. Find the Probability of Waiting at Least 4 Minutes**\r<BR>\n" );
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document.write( "* We need to find P(T ≥ 4), where T is the waiting time.\r<BR>\n" );
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document.write( "* **For an exponential distribution, the cumulative distribution function (CDF) is given by:**\r<BR>\n" );
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document.write( "   F(t) = 1 - e^(-λt) \r<BR>\n" );
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document.write( "* **To find P(T ≥ 4), we can use the complementary probability:**\r<BR>\n" );
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document.write( "   P(T ≥ 4) = 1 - P(T < 4) <BR>\n" );
document.write( "   P(T ≥ 4) = 1 - F(4) \r<BR>\n" );
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document.write( "* **Calculate F(4):**\r<BR>\n" );
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document.write( "   F(4) = 1 - e^(-1 * 4) <BR>\n" );
document.write( "   F(4) = 1 - e^(-4) <BR>\n" );
document.write( "   F(4) ≈ 1 - 0.0183 <BR>\n" );
document.write( "   F(4) ≈ 0.9817\r<BR>\n" );
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document.write( "* **Calculate P(T ≥ 4):**\r<BR>\n" );
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document.write( "   P(T ≥ 4) = 1 - F(4) <BR>\n" );
document.write( "   P(T ≥ 4) = 1 - 0.9817 <BR>\n" );
document.write( "   P(T ≥ 4) ≈ 0.0183\r<BR>\n" );
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document.write( "**Therefore, the probability that a customer will have to wait at least 4 minutes for a table is approximately 0.0183 (or 1.83%).**<BR>\n" );
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