document.write( "Question 1197515: An electronic system has four components labeled as 1, 2, 3, and 4. The system has to
\n" ); document.write( "be used during a given time period. The probability that component 𝑖 will fail during
\n" ); document.write( "that time period is Fi
\n" ); document.write( "for 𝑖 = 1, …, 4. Failures of the components are physically
\n" ); document.write( "independent of each other. A system failure occurs if component 1 fails or if at least
\n" ); document.write( "two of the other components fail. Specify an appropriate sample space and determine
\n" ); document.write( "the probability of a system failure. \n" ); document.write( "

Algebra.Com's Answer #830814 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "An electronic system has four components labeled as 1, 2, 3, and 4. The system has to
\n" ); document.write( "be used during a given time period. The probability that component 𝑖 will fail during
\n" ); document.write( "that time period is Fi
\n" ); document.write( "for 𝑖 = 1, …, 4. Failures of the components are physically
\n" ); document.write( "independent of each other. A system failure occurs if component 1 fails or if at least
\n" ); document.write( "two of the other components fail. Specify an appropriate sample space and determine
\n" ); document.write( "the probability of a system failure.
\n" ); document.write( "~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        If the final goal is to determine the probability for the whole system to fail, then
\n" ); document.write( "        it is much simpler to solve the problem without considering the appropriate sample space.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        Formula (1) below replaces considering sample space.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "If    is the probability to fail for i-th component,\r\n" );
document.write( "then the probability not to fail for this component is  = 1 - .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now the probability for the whole system to fail is\r\n" );
document.write( "\r\n" );
document.write( "    P = P(1st component failed) + P(1st is good, 2nd failed,  3rd failed,  4th is good)   + \r\n" );
document.write( "                                + P(1st is good, 2nd failed,  3rd is good, 4th failed)    + \r\n" );
document.write( "                                + P(1st is good, 2nd is good, 3rd failed,  4th failed)    + \r\n" );
document.write( "                                + P(1st is good, 2nd failed,  3rd failed,  4th failed).      (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is the same as\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    P =  +  +  = \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "      =  +  + .    ANSWER\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );