The probability is 1 in 4,010,000 that a single auto trip in the United States w

Question

The probability is 1 in 4,010,000 that a single auto trip in the United States will result in a fatality. Over a lifetime, an average U.S. driver takes 45,000 trips. (a) What is the probability of a fatal accident over a lifetime? (Hint: Assume independence.) (Round your answer to 4 decimal places.) Probability (b) Why might the assumption of independence be violated? Some drivers are better than others. An individual driver most likely has a higher chance of a fatal accident when he or she is a new driver than when he or she is more experienced. A fatal accident could be caused by someone other than the driver of the vehicle. (c) Why might a driver be tempted not to use a seat belt "just on this trip"? The small probability of a fatal accident means the consequences are not serious. The small probability of a fatal accident means that actually it will never happen. Humans tend to take risks when the probability of occurrence is very small.

Explanation / Answer

(a) Use the complement of the event:

P(a fatal accident over a lifetime) = 1 - P(NO fatal accident overa lifetime)

So we consider first:

P(NO fatal accident over a lifetime)

We are given:

P(ONE single auto trip in the United States resulting in afatality) =
1/4,010,000 = 0.000000249

Therefore

P(ONE single auto trip in the United States NOT resulting in a fatality) = 1-0.000000249 or 0.999999751

So

P(NO fatal accident over a lifetime) = P(NO fatal accident in the 45000 trips) =
P(No fatality on 1st trip AND
No fatality on 2nd trip AND
No fatality of 3rd trip AND
                          ...AND
No fatality of 44999th trip AND
No fatality on 45000th trip)


Since all those events are independent we can simply multiply alltheir probabilities together, which amounts to raising 0.999999751 to the 45000th power:

and 0.99999975145000 = 0.98885754
  
But that is the probability of the complement event.

P(a fatal accident over a lifetime) = 1 - P(NO fatal accident overa lifetime)

so

P(a fatal accident over a lifetime) = 1 - 0.98885754 =

0.011142459 or rounding, about 0.01114

b) When we assume independence, we are assuming that none of ourprevious driving experiences when there were no fatal accidents donot make it any less likely that we will have an accident. We(and the insurance companies) know this
is NOT TRUE because as a rule, we become better drivers when wehave had more experience. This is why the
insurance premiums are much lower for adult drivers than forteenage drivers. They know that the probability of
a fatal accident decreases as we get more experience driving -- and become more mature.

c) Well, he should not be so tempted, lest it become a habit. But if he if is really able to fail to buckle up only thisonce, then even assuming independence above,

P(ONE single auto trip in the United States resulting in afatality) =

1/4,010,000 = 0.000000249

and without independence it is an even smaller probability, and sohe is willing to take that small risk.

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