What is the probability of each type of ball going in a pipe?

[basantide]

Hi All

I am not a maths major (I have a engineering PhD) and am struggling with a probability problem that I need to include in one of my research problems.

Below is the a very general description of the type of the problem I am solving (not including the physics engineering equations etc behind it)

Lets say I have 2 types of balls. Ball A and Ball B. I have 400 balls of Ball A and 100 of Ball B.
I have have 50 pipes, P1 (25 pipes), P2 (20 pipes) and P3 (5 pipes) all with different characteristics (lets say different diameters)

What I want to calculate is the probability of each type of ball going in a pipe.

Right off the bat I know the there is a 80% probability of Ball A going in a pipe and 20% that of Ball B due to their total numbers (or frequency). But where I am getting confused is how to choose the pipes? OR what are the probability of the Ball X and P_X combined?

Would it be 80-20 ratio of balls in each type of pipe? Like, in case of P2 (20 pipes), would 16 pipes get Ball A and remaining 4 get Ball B?

Please help.

Basantide

 

[mathman]

To complete the question you need to describe the procedures involved, particularly ball selection and pipe selection.

 

[jaytech]

Excluding the procedures involved (not recommended), think of the probability of any ball entering a pipe. Now, once you have that figured out, factor in ball color. It should be a union of the two probabilistic sets:

P(Pipe) U P(Color).

 

[basantide]

There is no selection criteria except their respective number frequencies. The total number of each type ball remains constant (in my case these are concentration and they are the same throughout the process i.e 80-20 % is maintained throughout at least for now. In future I would have to take that into account as well but that's for later). Any ball can go in any pipe. The only thing I need to find the most likely combination in each case.

Hope that helps.

 

[basantide]

Thanks Jaytech. Can you explain me that in Layman's terms. I am far removed from the mathematical notations and symbols. Sorry.

 

[Stephen Tashi]

Below is the a very general description of the type of the problem I am solving (not including the physics engineering equations etc behind it)

If you are unfamiliar with probability theory, you will have a hard time stating problems involving probability with sufficient precision. I suggest you state the problem as a physics and engineering problem.

 

[basantide]

I understand a bit abot probability theory. This is what I am trying to solve:
I have a porous medium (with pore sizes and modeled as pipes) and a solution of various sized particles in water. Now when I start my experiment, a particle goes into a pore (i.e a pipe) and once it enters there I have equations to solve how it moves in that pore depending on its interactions with the pore surface. What I wanted on this forum was to figure out the most likely particle-pipe combination.

This is how I tried to get that:

80% Particle A (A) and 20% Particle B (B) (remains constant throughout the experiment as it is the concentration)
25 Pipe1 (name it X)
20 Pipe2 (Y)
5 Pipe3 (Z)

So if run this experiment for infinite time, I should get the following probabilities or likely events:

P (AX) = 0.8*0.5
P(BX) = 0.2*0.5
P(AY) = 0.8*0.4

etc

Is this right?

 

[mathman]

The probabilities assume independence of particle size and pipe characteristics. Also the probability of entering a pipe is assumed dependent only on the number of pipes, but not size.

Whether or not these assumptions are correct is a question of physics, not mathematics.

 

[Stephen Tashi]

I should get the following probabilities or likely events:

P (AX) = 0.8*0.5
P(BX) = 0.2*0.5
P(AY) = 0.8*0.4

etc

Is this right?

You haven't defined the space of possible events. That' should be the first step in dealing with probabilities.

My best guess for the experiment that defines the space of events is: Pick a pipe at random and pick a ball at random that went into that pipe. Then the event "AX" denotes the outcome that the type of the pipe picked was X and the type of the ball picked was A.

If you want "AX" to mean something about the probability of a ball of type A going into a pipe X then you need to define that experiment. For example, is it an experiment where a ball of type A might not go into any of the pipes?

If you are trying to do a calculation about concentrations, then don't confuse concentrations with probabilities.

 

[basantide]

The experiment is such that any ball can go in any pipe. What I am doing with modeling is is calculating the flow of water (since the balls are in water i.e it is solution) through the pipes and balls (impurities in water) act as some type of resistance. The physics of that resistance is captured once the ball is in the pipe. Right now I just randomly select a ball and a pipe as Stephen suggested above. But I am trying to work in the probabilities AX, BX, By etc just to have some basis to select ball/pipe combination instead of it being random.

 

[FactChecker]

Your description doesn't give you any way to determine what combination of ball and pipe are occurring. In an extreme example where one pipe diameter is 1000 times as large as the ball and another is smaller than the ball, your description wouldn't allow you to take that into account at all. You need to do that.

 

[basantide]

OK. Let's say one of the particles is (say, Particle B) is larger in diameter than the size of any of the pipes, thus it won't enter the pies, but it will sit on the pipe entracnce and block particle A going through that particular tube. How would the probability problem work out in this case?

Thanks for all your suggestions so far.

 

[Stephen Tashi]

How would the probability problem work out in this case?

You expect too much of mathematics. Mathematics requires many "givens" in order to reach a conclusion. You should state the actual physical problem.

If you don't see how to apply mathematics then its not likely that you yourself can successfully state exactly the abstract details from the physical situation that provide enough "givens". A useful model your problem in terms of balls going into pipes is something that can only be created after the actual physical situation is known. It also requires that you define what your goal is in the the analysis - by that I mean what physical result you are concerned with, not what abstract probability question.

 

[basantide]

Here is the actual problem I am trying to solve. We have two sizes of particles in a water solution at different concentrations. We push this solution over a membrane that has pores (or tubes) of given diameters and total numbers. What we experimentally is to record how much water are we getting on the other side of the membrane.
The physical mechanism in this situation is :

1. The smaller particles (particle diameter smaller than the pore/tube diameter) will enter that pore and due to interactions (electrostatic, diffusion etc) with the surface of the pore will get attached to the pore and form a resistance to water flow (since now due to particle attachment, the pore size will start to reduce).
2. The bigger particles (particle diameter larger than the pore/tube diameter) will get to the pore opening but due to its size will block that pore and hence no water will flow through that pore.

Also, all the pores are exposed to the solution at the same time and hence we can have a situation where smaller particles go through for some time and then the bigger particle will block that pore. But to simplify the modeling, I consider only one size of particles can through the pore i.e. either smaller or bigger particle throughout the modeling "experiment".

With respect to modeling, I have captured the particle-pore surface interaction "inside" the pore to see how many particles are attached and their resistance to water flow.

What I am struggling with is how to assign what particle will go through what pore given the sizes and the total numbers (or concentrations) for both the particles and pores.

Hope this makes it a little clearer.

 

[Stephen Tashi]

We have two sizes of particles in a water solution at different concentrations.

We push this solution over a membrane that has pores (or tubes) of given diameters and total numbers.

I visualize this a cup with a membrane over it and the solution being poured over membrane with some of the solution allowed to run off the sides. A different possibility is that the membrane could be used to cover the end of the outflow of a pipe so that none of the solution could bypass it.

What we experimentally is to record how much water are we getting on the other side of the membrane.

Do you measure the total water at the end of a fixed time? Or do you measure the water that has passed through as a function of time?

Or do you start with a fixed volume of solution and let the experiment last for however long it takes for that volume to be poured over the membrane?

Do you measure the new concentrations of the particles in the water that has passed through the pores? Are we assuming, for simplicity, that all particles are captured inside the pores?

What I am struggling with is how to assign what particle will go through what pore given the sizes and the total numbers (or concentrations) for both the particles and pores.

If I imagine the particles as tiny and numerous relative to the pores and the rate of flow through pores relatively fast compared to the motions of the individual particles then concentration of particles entering each size pore would be the same and the number of particles of each type that enter a pore (in a given time) would depend on the flow rate of the solution through the pore.

Does the situation differ from that simplistic picture?

 

[basantide]

Hi Stephen

1. The membrane is inside a leak free module (fittings), so nothing runs off the sides.
2. We measure water flow as a function of time. A fixed volume of solution is used and the experiment is run for however long it takes for most of that volume (80 to 90%) to be poured over the membrane.
3. Yeah we do not measure the concentration of particles in the water that has passed. We, as you said, for simplicity assume all the particles that go through are captured in the pores.
4. "If I imagine the particles as tiny and numerous relative to the pores and the rate of flow through pores relatively fast compared to the motions of the individual particles then concentration of particles entering each size pore would be the same and the number of particles of each type that enter a pore (in a given time) would depend on the flow rate of the solution through the pore." this is right. Also the concentration of particles in the feed solution stays constant throughout the experiment as the solution is well mixed.

 

[FactChecker]

This sounds like a problem that might require a Monte Carlo simulation to study. The flow of water through different size pipes draws different diameter balls along. The physics of that may have to be simulated.

 

[Stephen Tashi]

One way to model the situation in discrete time steps of dt is that the total flow of volume dV(t) is equal to the sum of the flows dv(t) through each of the ith pore. The number of particles of a given type that enter the ith pore is a determined by dt times the rate of flow through the pore at time t. You'd rely on hydrodynamics to tell you the flow rates. The model wouldn't involve probability.

If a single particle entering a pore can have a big effect on the flow rate through the pore then it's natural to seek a model that deals with individual particles. To modify the above model, we can make dt so small that the volume dv(t) flowing through the ith pore contains at most 1 particle. I suppose this would lead to assuming the particles have a poisson distribution in the volume of the fluid.

 

[basantide]

Hi Stephan

Thank you for your help. But my question I believe still stands. I can do all you said in the above reply. But my original question was how do I choose which particle type goes to the ith pore? Becuase the type of particle has a effect on the water flow.

 

[Stephen Tashi]

But my question I believe still stands.

You haven't specified enough "givens" to answer a mathematical question. The actual physical process you describe takes place in space and in time. You are asking questions like "What is the probability that a ball of type A goes into a pore of type X" that can only be answered by pencil and paper calculations if we assume a completely different physical situation - such as someone picking a ball at random from collection of balls and then picking a tube at random from a collection of tubes.

If you use the deterministic model model of flow I described, there is no probability involved.

If it matters which type of particle enters a pore first then you must simulate a process that takes place in time and since particles can enter different pores during the same time interval you must simulate a process that takes place in space.

Resorting to some mind reading I'll guess the following:

You hope to avoid simulating the spatial distribution of particles. You want the simulation to work in steps where each step consists of picking a random particle and then picking a random pore for it to enter. In other words, the particles are approaching the membrane in a single file and the particle at the head of the line is sent through a pore before the particle behind it.

If the rates of flow though various pores change, then you must not only simulate a line of particles approaching the membrane, you must also simulate how the particle entries are spaced apart in time.

I don't see any way of guessing how to implement the above type of simulation. I suggest thinking about a model that has more geometric complexity.

We could think of an essentially 1-D model. Suppose the current rate of flow through the ith pore is dv(t) and the total rate of flow through all pores is dV(t). In the finite time delta_t, approximately a volume of fluid delta_V = delta_t dV(t) passes through the pores. Imagine the volume delta_V poured in a set of tiny cups of equal volume,. The cups are so small that the probability that a cup contains two particles is negligible. (Think of the dimensions of the cups as being small relative to how far apart particles are typically spaced in the solution.) For a given pore, we imagine that during time delta_t, a certain number of these small cups are poured, 1 by 1 into to the pore. The total number of these cups is chosen to make the total volume entering the pore delta_t dV(t).

We can make stochastic decision about whether a given cup contains a particle. If the solution contains on the average k particles per cc then we can compute the average number of particles r in the volume of a tiny cup. (r < 1 since the cups are small). Let the probability that a cup contains a particle be r.

In this simulation, at each time step delta_t we form a line of cups in front of each pore. For each cup, we make a random determination whether a particle is placed in the cup. Then it's up to a more detailed simulation to simulate what happens as the line of cups enters the pore..

This is only a conceptual model and it probably needs refinement, both physically and mathematically.

 

[FactChecker]

Hi Stephan

Thank you for your help. But my question I believe still stands. I can do all you said in the above reply. But my original question was how do I choose which particle type goes to the ith pore? Becuase the type of particle has a effect on the water flow.

That is what we are trying to arrive at. Sorry, but the probability may be too hard to calculate directly. If the basic physics facts are known, you can do many Monte Carlo simulated "experiments" and keep track of which balls go into which pipes in the simulation. After enough simulations you can estimate the probabilities. That can be done even when there are hundreds of physics equations that apply. Monte Carlo simulations can be used on problems that were directly calculating probabilities is inconceivable.

 

[AlephZero]

I think you are attacking this from the wrong end, possibly by having too much blind faith in mathematics (as others have said).

I think a valid way to "solve" the problem is
1. Do some experiments to find out what actually happens in real life.
2. Then try to create a mathematical model (using probabilities if you want) that produces the same results.

The reason this isn't going anywhere useful is because you are trying to skip step 1.

At the minimum, you might decide that the physics of this is similar to osmosis, for example (but from the limited amount of description in the thread, that is just a guess) and then find out what has already been done to model osmosis, and see if it gives you what you want. But without doing some experiments, how do you know if the math is giving the "right answers"?