Probabilities: A freight train consisting of C initially empty hopper cars passes thr

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SimsPhilly

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  1. A freight train consisting of C initially empty hopper cars passes through S stations.
    1. (a) Suppose that at each station, a random car is selected (uniformly across cars) and a package is placed in it, unless this car is already full to capacity, in which case the package falls on the ground and is lost. Each car has a capacity of M packages.
      1. Express the probability that none of the stations packages fall on the ground and get lost this way as a function of C, S, and M.
      2. What is the probability that at least one of the C cars is still empty after all S stations are visited?


 
SimsPhilly said:

  1. A freight train consisting of C initially empty hopper cars passes through S stations.
    1. (a) Suppose that at each station, a random car is selected (uniformly across cars) and a package is placed in it, unless this car is already full to capacity, in which case the package falls on the ground and is lost. Each car has a capacity of M packages.
      1. Express the probability that none of the stations packages fall on the ground and get lost this way as a function of C, S, and M.
      2. What is the probability that at least one of the C cars is still empty after all S stations are visited?
Click to expand...
Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

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Subhotosh Khan said:
Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/for
Click to expand...


Thank you for your feedback.

I'm actually stuck at the beginning and I have the lectures notes with me. I can't figure out in what situation I am.
Not sure if I should be looking at the simple probability laws or at the counting principles (combination, permutation,..)
I can't figure out what is my space and the way I can express the probabilities

Thank you again

Sims
 
SimsPhilly said:

  1. A freight train consisting of C initially empty hopper cars passes through S stations.
    1. (a) Suppose that at each station, a random car is selected (uniformly across cars) and a package is placed in it, unless this car is already full to capacity, in which case the package falls on the ground and is lost. Each car has a capacity of M packages.
      1. Express the probability that none of the stations packages fall on the ground and get lost this way as a function of C, S, and M.
      2. What is the probability that at least one of the C cars is still empty after all S stations are visited?
Click to expand...

SimsPhilly said:
I'm actually stuck at the beginning and I have the lectures notes with me. I can't figure out in what situation I am.
Not sure if I should be looking at the simple probability laws or at the counting principles (combination, permutation,..)
I can't figure out what is my space and the way I can express the probabilities
Click to expand...

Thanks for telling us some of your context. It may also be important to tell us what you have learned. Are you just at the start of studying basic probability, or are you in a more advanced topic? What is the most recent topic, supposing that this may be meant to test something that was just taught?

Looking only at the first part, I'd try paraphrasing the problem:
There are S trials, each consisting of choosing one of C containers and adding 1 to its current content; any time the content is already M counts as a failure. We want the probability that there are no failures (that is, each of the S trials succeeds).
​

Now I try simplifying that further:
In each of S trials, one of C objects is randomly chosen. We want the probability that no one object is chosen more than M times.
​

I might try modeling it differently, perhaps like this:
We have a table containing S rows of C cells. In each row, we randomly choose one cell to mark. What is the probability that no column contains more than M marks?
​

Also, at some point along the way, I noticed that if C<=M, the probability is 1. So there will probably be at least two cases (making the answer a piecewise-defined function).

Something like that may make it easier to see a way to solve the problem. (And there are probably many ways to solve it, so I can't point to one way as the one you should discover.) I'm inclined toward a combination or permutation approach, though.
 
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