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RES/341 Week 4 Answer Guide - Statistics

5.62 -A certain airplane has two independent alternators to provide electrical power. The probability that a given alternator will fail on a 1-hour flight is .02. What is the probability that (a) both will fail? (b) Neither will fail? (c) One or the other will fail? Show all steps carefully.
(a) Both Fail
N = 2, p = .02, a = alternator 1, b = alternator 2
Pa Ç Pb =
(.02)(.02) = 0.0004
The probability that both will fail is 0.0004 or 0.04%


 


(b) Neither Will Fail
1-( Pa + Pb) + (Pa Ç Pb) =
(1 - 0.04) + 0.0004 = 0.9604
The probability that neither will fail is 0.9604 or 96.04%

(c) One or the other will fail
Pa + Pb - (Pa Ç Pb)=
0.02 + 0.02 – 0.0004 = 0.0396
The probability that one or the other will fail is 0.0396 or 3.96%
5.70 - * 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 seat belt “just on this trip”?

(a) What is the probability of a fatal accident over a lifetime?
N = 2, p = 0.00000025, a = fatal accident, b = no fatal accident
Pa = 1 – Pb 
Pa = 1 – 0.00000025
Pa = 0.99999975 ^ 50000 = 0.987577799
Pa = 1 – 0.987577799
Pa = 0.012422201

The probability of a fatal accident over a lifetime is 0.012422201 or 1.24%.

I came to this conclusion by first using the complement rule. There are two possible outcomes from each trip which are fatal accident (a), and non-fatal accident (b). From here, I raised the result to the 50,000th power for each possible trip. This result was subtracted from 1 to get the final probability of 1.24%.
Why might the assumption of independence be violated?
The assumption of independence is violated because drivers do not have identical experiences on each trip. The length and difficulty of each trip can vary significantly. For example, a trip to the grocery store would assumingly be much safer than a 1,000 mile road trip. The time of day will also vary, and trips during rush hour may be less safe due to the crowded road conditions. Also, the experience and skill of the driver will normally increase with age, which would make an accident less likely over time.

(b) Why might a driver be tempted not to use a seat belt “just on this trip”?

A driver might be tempted not to use a seatbelt on just one trip if it is short in short in duration. The driver may assume that the short duration makes it less likely for an accident to occur. Not wearing a seatbelt would assumingly increase the probability of a fatality in the event of an accident. This would violate the assumed independence.

 

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