1. ## Probability

I need some help PLEASE! These are two of the problems that I need to answer for my Statistics class and I have no idea what to do or where to begin. If you could help me I would greatly appreciate it!

#1. 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.

#2. 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”?

2. Originally Posted by foofergutierrez
I need some help PLEASE! These are two of the problems that I need to answer for my Statistics class and I have no idea what to do or where to begin. If you could help me I would greatly appreciate it!

#1. 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.

#2. 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”?
the fact that we have independent events and two possible outcomes are the key here. we can use Bernoulli trials, that is, the Binomial distribution to answer these problems.

assuming all the conditions are fulfilled (independent trials, two possible outcomes per trial that can be thought of as success or failure ...), we have the following by the method of Bernoulli trials.

the probability of $k$ successes in $n$ trials, where the probability of success is $p$ and the probabilty of failure is $q = 1 - p$ is given by:

$P(k) = {n \choose k} p^kq^{n - k}$

so for question (1), you have: $n = 2,~p = 0.02$

for (a) you want $P(2)$

for (b) you want $P(0)$

for (c) you want $P(1)$

for question one, it was also possible to reason through it.

part (a) is asking the probability that alternator 1 will fail AND alternator 2 will fail. since we have independent trials, we can just multiply the probabilities

part (b) is asking the probability that alternator 1 does not fail AND the probability that alternator 2 does not fail, again, we just multiply the respective probabilities

part (c) is asking for the probability the alternator 1 fails but 2 doesn't OR the probability that alternator 1 does not fail, but 2 does. we can ADD the probabilities for each case here, since we have OR as opposed to AND

for question (2), you have: $n = 50000,~p = 1/4000000$

for the first part of part (a), you simply want $P(1)$. for the second part, you could say the car accidents might not be independent, but one accident can cause another...what other way can you think of that the accidents may not be independent?

for part (b), answer part (a) and you should have it

3. Originally Posted by Jhevon
for the first part of part (a), you simply want $P(1)$.
I think it’s more likely you want $\mathrm{P}(\geq1)$.

4. Originally Posted by foofergutierrez
#2. 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?
The probability of having one (or more) fatal accidents in a lifetime is:

(that is 1 minus y=the probability that you survive each and every journey)

Why the independence assumption might be violated is psychology
not statistics, but a long period without an accident could give rise to
overconfidence and hence less care when driving, and so the probability
of surviving a journey could go down as more journeys are survived.

(b) Why might a driver be tempted not to use a seat belt “just on this trip”?
Can't imagine, stupidity probably

RonL

5. ## Thanks!!

I would just like to say thank you all because your answers helped me to figure it out and hopefully I'll remember it. The book I have is not very discriptive so that is why I asked for help and others in the same class are having issues with the same problems. I am sure I will be back on with another question or two so be prepared. . . and thanks again

6. Originally Posted by CaptainBlack

Can't imagine, stupidity probably

more likely stupidity in disguise. i guess we should assume the driver knows the answer to the previous question, in which he/she might be tempted to say..."Oh, the chances of being in a fatal accident is just one percent, it won't happen to me." this is stupid, not only for the obvious reason, but because the reason some of the accidents aren't fatal might be because the passengers were wearing seatbelts...