document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192426>Question 1192426</A>: <I><SPAN STYLE=\"word-break: break-all;\">  The studies of a single machine tool system showed that the machine <BR>\n" );
document.write( "operates before breaking down is exponentially distributed with a mean of <BR>\n" );
document.write( "10 hours.<BR>\n" );
document.write( "a) Find the probability that the machine operates at least 12 hours before <BR>\n" );
document.write( "breaking down.<BR>\n" );
document.write( "b) Find the probability that the machine operates at least 14 hours but <BR>\n" );
document.write( "less than 20s hours before breaking down. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848990 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=848990\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Probability of operating at least 12 hours**\r<BR>\n" );
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document.write( "* **Given:**<BR>\n" );
document.write( "    * Mean time to failure (μ) = 10 hours <BR>\n" );
document.write( "    * Rate parameter (λ) = 1/μ = 1/10 = 0.1 per hour \r<BR>\n" );
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document.write( "* **Probability of operating at least 12 hours:**<BR>\n" );
document.write( "    * P(X ≥ 12) = 1 - P(X < 12) <BR>\n" );
document.write( "    * P(X ≥ 12) = 1 - (1 - e^(-λx)) <BR>\n" );
document.write( "    * P(X ≥ 12) = 1 - (1 - e^(-0.1 * 12)) <BR>\n" );
document.write( "    * P(X ≥ 12) = 1 - (1 - e^(-1.2)) <BR>\n" );
document.write( "    * P(X ≥ 12) = e^(-1.2) <BR>\n" );
document.write( "    * P(X ≥ 12) ≈ 0.3012\r<BR>\n" );
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document.write( "**b) Probability of operating between 14 and 20 hours**\r<BR>\n" );
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document.write( "* **Probability of operating at least 14 hours:**<BR>\n" );
document.write( "    * P(X ≥ 14) = e^(-0.1 * 14) = e^(-1.4) ≈ 0.2466\r<BR>\n" );
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document.write( "* **Probability of operating at least 20 hours:**<BR>\n" );
document.write( "    * P(X ≥ 20) = e^(-0.1 * 20) = e^(-2) ≈ 0.1353\r<BR>\n" );
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document.write( "* **Probability of operating between 14 and 20 hours:**<BR>\n" );
document.write( "    * P(14 ≤ X < 20) = P(X ≥ 14) - P(X ≥ 20) <BR>\n" );
document.write( "    * P(14 ≤ X < 20) = 0.2466 - 0.1353 <BR>\n" );
document.write( "    * P(14 ≤ X < 20) ≈ 0.1113\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* a) The probability that the machine operates at least 12 hours before breaking down is approximately 0.3012.<BR>\n" );
document.write( "* b) The probability that the machine operates at least 14 hours but less than 20 hours before breaking down is approximately 0.1113.<BR>\n" );
document.write( " </SPAN>\n" );
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