document.write( "Question 125800: The probability is 1 in 4,000,000 that a single auto trip in the United States will result in a fatality. Over a lifetime , an average U.S. driver takes 50,000 trips. (a) What is the probability of a fatal accident over a lifetime? Explain your reasoning carefully. Hint: Assume independent events. why might the assumption of independence be violated? (b) why might a driver be tempted not to use a seatbelt \"just on this trip\"? \n" ); document.write( "
Algebra.Com's Answer #92186 by Edwin McCravy(20056)\"\" \"About 
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The probability is 1 in 4,000,000 that a single auto trip in the United States will result in a fatality. Over a lifetime , an average U.S. driver takes 50,000 trips. (a) What is the probability of a fatal accident over a lifetime? Explain your reasoning carefully. Hint: Assume independent events.
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document.write( "We use the complement event:\r\n" );
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document.write( "P(a fatal accident over a lifetime) = 1 - P(NO fatal accident over a lifetime) \r\n" );
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document.write( "So we consider first:\r\n" );
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document.write( "P(NO fatal accident over a lifetime) \r\n" );
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document.write( "We are given:\r\n" );
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document.write( "P(ONE single auto trip in the United States resulting in a fatality) =\r\n" );
document.write( "\"1%2F4000000\" = 0.00000025\r\n" );
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document.write( "Therefore \r\n" );
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document.write( "P(ONE single auto trip in the United States NOT resulting in a fatality) =\r\n" );
document.write( "1-0.00000025 or 0.99999975\r\n" );
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document.write( "So\r\n" );
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document.write( "P(NO fatal accident over a lifetime) = P(NO fatal accident in the 50000 trips) = \r\n" );
document.write( "P(No fatality on 1st trip AND \r\n" );
document.write( "  No fatality on 2nd trip AND \r\n" );
document.write( "  No fatality of 3rd trip AND\r\n" );
document.write( "                           ...AND\r\n" );
document.write( "  No fatality of 49999th trip AND \r\n" );
document.write( "  No fatality on 50000th trip)\r\n" );
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document.write( "Since all those events are independent we can simply multiply all\r\n" );
document.write( "their probabilities together, which amounts to raising  0.99999975 to \r\n" );
document.write( "the 50000th power:\r\n" );
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document.write( "and 0.9999997550000 = 0.987577799\r\n" );
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document.write( "But that is the probability of the complement event.\r\n" );
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document.write( "P(a fatal accident over a lifetime) = 1 - P(NO fatal accident over a lifetime)\r\n" );
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document.write( "so\r\n" );
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document.write( "P(a fatal accident over a lifetime) = 1 - 0.987577799 =\r\n" );
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document.write( "0.012422201 or rounding, about 0.0124\r\n" );
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document.write( "That's scary, isn't it?  However there is relief in\r\n" );
document.write( "the next question:\r\n" );
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\n" ); document.write( "why might the assumption of independence be violated?
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document.write( "When we assume independence, we are assuming that none of \r\n" );
document.write( "our previous driving experiences when there were no fatal \r\n" );
document.write( "accidents do not make it any less likely that we will have\r\n" );
document.write( "an accident.  We (and the insurance companies) know this \r\n" );
document.write( "is NOT TRUE because as a rule, we become better drivers \r\n" );
document.write( "when we have had more experience.  This is why the \r\n" );
document.write( "insurance premiums are much lower for adult drivers than\r\n" );
document.write( "for teenage drivers.  They know that the probability of\r\n" );
document.write( "a fatal accident decreases as we get more experience\r\n" );
document.write( "driveing -- and become more mature.\r\n" );
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\n" ); document.write( "(b) why might a driver be tempted not to use a seatbelt \"just on this trip\"?
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document.write( "Well, he should not be so tempted, lest it become a habit. But if\r\n" );
document.write( "he if is really able to fail to buckle up only this once, then even\r\n" );
document.write( "assuming independence above, \r\n" );
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document.write( "P(ONE single auto trip in the United States resulting in a fatality) =\r\n" );
document.write( "\"1%2F4000000\" = 0.00000025 \r\n" );
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document.write( "and without independence it is an even smaller probability, and so he is \r\n" );
document.write( "willing to take that small risk.\r\n" );
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document.write( "Edwin
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