document.write( "Question 376519: * The probability of a successful assembly of a product in a manufacturing process is 0.8.\r
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\n" ); document.write( "\n" ); document.write( "ii) What is the probability that the first successful assembly requires at least four trials?\r
\n" ); document.write( "\n" ); document.write( "iii) What is the probability that the first successful assembly requires at most four trials?iv) What assumptions did you make, if any, to solve this problem?\r
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Algebra.Com's Answer #267879 by Edwin McCravy(20060)\"\" \"About 
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* The probability of a successful assembly of a product in a manufacturing process is 0.8.\r
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document.write( "Probability of a success is 0.8 \r\n" );
document.write( "Probability of a failure is 1 - 0.8 = 0.2\r\n" );
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document.write( "P(failure 1st AND failure 2nd AND failure 3rd AND success 4th) =\r\n" );
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document.write( "P(failure 1st)*P(failure 2nd)*P(failure 3rd)*P(success 4th) =\r\n" );
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document.write( "\"%280.2%29%280.2%29%280.2%29%280.8%29+=++%28.02%29%5E3%280.8%29+=+0.0064\"\r\n" );
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\n" ); document.write( "ii) What is the probability that the first successful assembly requires at least four trials?
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document.write( "The probability that the first successful attempt will be the nth trial is\r\n" );
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document.write( "\"%280.8%29%5E%28n-1%29%2A%280.2%29\"\r\n" );
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document.write( "That is the nth term of an infinite geometric series with first term\r\n" );
document.write( "\"a%5B1%5D=0.0064\" and common ratio r = 0.8.\r\n" );
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document.write( "That sum is given by\r\n" );
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document.write( "\"S%5Binfinity%5D\"\"%22%22=%22%22\"\"a%5B1%5D%2F%281-r%29\"\r\n" );
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document.write( "\"S%5Binfinity%5D\"\"%22%22=%22%22\"\"0.0064%2F%281-0.8%29=0.0064%2F0.2=0.032\"\r\n" );
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\n" ); document.write( "iii) What is the probability that the first successful assembly requires at most four trials?
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document.write( "The probability that the first successful attempt will be the nth trial is\r\n" );
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document.write( "\"%280.8%29%5E%28n-1%29%2A%280.2%29\"\r\n" );
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document.write( "So\r\n" );
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document.write( "The probability that the first successful attempt will be the 1st trial is\r\n" );
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document.write( "\"%280.8%29%5E%281-1%29%2A%280.2%29=%280.8%29%5E0%2A%280.2%29=1%2A0.2=0.2\"\r\n" );
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document.write( "The probability that the first successful attempt will be the 2nd trial is\r\n" );
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document.write( "\"%280.8%29%5E%282-1%29%2A%280.2%29=%280.8%29%5E1%2A%280.2%29=%280.8%29%280.2%29=0.16\"\r\n" );
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document.write( "The probability that the first successful attempt will be the 3rd trial is\r\n" );
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document.write( "\"%280.8%29%5E%283-1%29%2A%280.2%29=%280.8%29%5E2%2A%280.2%29=%280.64%29%280.2%29=0.128\"\r\n" );
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document.write( "The probability that the first successful attempt will be the 4th trial is\r\n" );
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document.write( "\"%280.8%29%5E%284-1%29%2A%280.2%29=%280.8%29%5E3%2A%280.2%29=%280.512%29%280.2%29=0.1024\"\r\n" );
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document.write( "The required probability is the sum of those\r\n" );
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document.write( "0.2 + 0.16 + 0.128 + 0.1024 = 0.5904\r\n" );
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\n" ); document.write( "iv) What assumptions did you make, if any, to solve this problem?
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document.write( "The assumption that the trials are independent, that is, the assumption\r\n" );
document.write( "that previous successful or failing trials do not affect the probability \r\n" );
document.write( "of present or future trials being successful or failing.\r\n" );
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document.write( "Edwin
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