document.write( "Question 1118546: In a home theater system, the probability that the video components need repair within 1 year is 0.03, the probability that the electronic components need repair within 1 year is 0.007, and the probability that the audio components need repair within 1 year is 0.004. Assuming that the events are independent, find the following probabilities. (Round your answers to four decimal places.)
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Algebra.Com's Answer #734161 by ikleyn(52879) $\"\"$ $\"About$

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\n" ); document.write( "In a home theater system, the probability that the video components need repair within 1 year is 0.03,
\n" ); document.write( "he probability that the electronic components need repair within 1 year is 0.007,
\n" ); document.write( "and the probability that the audio components need repair within 1 year is 0.004.
\n" ); document.write( "Assuming that the events are independent, find the following probabilities. (Round your answers to four decimal places.)\r
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document.write( "(a)  At least one of these components will need repair within 1 year.\r\n" );
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document.write( "     According to the condition, the probability that\r\n" );
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document.write( "        V-component will not need a repair within 1 year is  (1-0.03),\r\n" );
document.write( "        E-component will not need a repair within 1 year is  (1-0.007),\r\n" );
document.write( "        A-component will not need a repair within 1 year is  (1-0.004).\r\n" );
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document.write( "     Hence, the probability that NO ONE of the three components will not need a repair is the product  (1-0.03)*(1-0.007)*(1-0.004).\r\n" );
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document.write( "     Then the probability under the question has the complementary value\r\n" );
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document.write( "         1 - (1-0.03)*(1-0.007)*(1-0.004) = 0.0406 = 4.06%.\r\n" );
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document.write( "(b)  Exactly one of these components will need repair within 1 year.\r\n" );
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document.write( "         (1-0.03)*(1-0.007)*0.004 + (1-0.03)*0.007*(1-0.004) + 0.03*(1-0.007)*(1-0.004) = 0.04030 = 4.030%.\r\n" );
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document.write( "     Explanation. \r\n" );
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document.write( "         The probability that only A-component will need a repair within 1 year is  (1-0.03)*(1-0.007)*0.004.\r\n" );
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document.write( "         The probability that only E-component will need a repair within 1 year is  (1-0.03)*0.007*(1-0.004).\r\n" );
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document.write( "         The probability that only V-component will need a repair within 1 year is  0.03*(1-0.007)*(1-0.004).\r\n" );
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document.write( "         The probability that exactly one of these three independent events will happen within 1 year is the sum \r\n" );
document.write( "         of these particular probabilities, which coincides with my answer. \r\n" );
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