The likelihood of an ambulance to be down for repairs is 5%. I have an
inventory of 20 ambulances. What is the probability that there will be
5 ambulances down for repair at any point of time?
Can you please explain how this question will be solved, will you
treat it is a binomial distribution?

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Clarification of Question by tears_in_heaven-ga on 03 May 2005 18:11 PDT

Thanks or the clarification, my answer doesn't agree with yours
though. Also whats the likelihood that you would need at least 7
vehicles to provide coverage for the area under consideration.

http://www.stat.tamu.edu/stat30x/notes/node66.html

That page will walk you through how to work the problem as a binomial
distribution.  I'd do it for you, but then there wouldn't be any
learning going on.

The final answer will be .2244646%, so you can check your work.  Best
of luck to you!  If you get stuck on any part of it feel free to ask.

Hmmm, sorry that didn't work out for you yet.  I worked the problem
another way to make sure the answer was right and I got the same
answer.
The only things I know to suggest (since I can't see your work) is 

1) Assign all the variables before working through the problem.  
2) Be sure that the probability of success is right (ie if you're
looking for 5 ambulances being repaired then 5% is your "probability
of success" (as opposed to 95% which it would be if you were looking
for 15 ambulances that are in working order).
3) Remember when calculating that 5% = .05 and 95% = .95

I hope that helps!  The answer for 7 ambulances under repair is .0031089%.

actually i get the same answers. Let me explain the context to this
question. I work for the local government which provides emergency
services in our city. The Emergency Services department is advocating
building 20 parking bays for our fleet of ambulances and having 100%
redundancy i.e if there are 20 ambulances being used during peak hours
there would be equivalent amount of ambulances parked in the bays to
provide redundancies. Obviously this has implications in terms of
extra cost of buying vehicles and capital requirements to build
additional bays. I'm trying to figure out with say 85%,90%,95%
availability factor for such ambulances what is the likelihood that 5
or more, 6 or more, 7 or more etc. ambulances would be needed and on
that basis how many ambulances should be kept in reserve.

I guess my second question related to this would be whether it is
reasonable to assume "availability" of amublances as a
percentage....i.e 85%. The reason i ask is because the binomial
distribution is used for binary events which have a yes or no result,
e.g flipping a coin results in heads or tails. However the
avialability of each ambulance will (or should) follow a normal
distribution curve (or some other curve), so the actual availability
may be .85 with a standard deviation of .02 etc....in this case is it
reasonable to still model the problem as a binomial distribution to
get a simplistic answer?

I guess the most appropriate way of modeling this would by via
simulations models? Do you agree?

I think a binomial could work for you, but a simple reality check is
the best way to go here.  You know from experience how many ambulances
get used and how often they are in for repair.  Take the most you have
ever had a need for in the past and there is a safe number to go with.
 If you're willing to take on a slightly greater risk of not having
enough then consider having 1 less and take a look at how many days in
the last few years there would have been a shortfall with that number.
 If you want less risk then add 1 or 2 ambulances to that number
(keeping in mind that they are useless if you don't have the people to
man them).
If the past data on ambulance use isn't available then you should
definately call to several different communities to find out how many
ambulances they use and ask for any statistics they have (keeping in
mind the number of people they service).

I asked the same question in another thread, unfortunately the answers
were not very forthcoming...
http://answers.google.com/answers/threadview?id=514959

I agree, its worth asking and phoning other communities, but it seems
that most communities do not measure redundancy in fleets in such a
manner. This kind of information is hard to find.

i think some kind of theoretical basis is important, i'm not too sure
if we have the data to indicate down time.

Anyways thanks for your help.