Quick question on probability, odds etc.

[saor]

Assuming the following experiment is performed only once:

I have a bucket of 99 green marbles and 1 red marble.
Bucket is shaken.
I reach in and pick the red marble.

Actually, the experiment could be done with any number of green marbles where x is any natural number:

I have a bucket of x green marbles and 1 red marble.
Bucket is shaken.
I reach in and pick the red marble.

Does the perceived unlikeliness of the red marble being selected from a bucket of 10 million green marbles have any objective formal expression, or is it's perceived unlikeliness merely a subjective quality we ascribe to the result? I'm leaning toward the latter - that there's nothing special about the red marble being chosen despite the size of the experiment. But, it feels like there's a real sense in which it is unlikely .

Thoughts?

 

[Xarog]

Assuming the following experiment is performed only once:

I have a bucket of 99 green marbles and 1 red marble.
Bucket is shaken.
I reach in and pick the red marble.

Actually, the experiment could be done with any number of green marbles where x is any natural number:

I have a bucket of x green marbles and 1 red marble.
Bucket is shaken.
I reach in and pick the red marble.

Does the perceived unlikeliness of the red marble being selected from a bucket of 10 million green marbles have any objective formal expression, or is it's perceived unlikeliness merely a subjective quality we ascribe to the result? I'm leaning toward the latter - that there's nothing special about the red marble being chosen despite the size of the experiment. But, it feels like there's a real sense in which it is unlikely .

Thoughts?

Statistical probability is more like a measurement of all the variables that could end up affecting the draw. It takes a simplified (frequently over-simplified) version of events and predicts how often those variables will end in whatever result you are trying to measure.

So imo it's quite subjective.

 

[saor]

Statistical probability is more like a measurement of all the variables that could end up affecting the draw.

I had in mind a simple system with parts (marbles) that aren't weighted or loaded in any way. For the sake of simplicity, would you agree with both these statements:

1.) In a bucket of 100 green marbles - each has a 1:100 chance of being selected.
2.) In a bucket of 99 green marbles and one red marble - each has a 1:100 chance of being selected.

No matter how big the bucket of marbles becomes, the probability of the unique red marble being selected will have the same probability as any other marble being selected. Again - it feels like it should be more improbable that the red marble be chosen, but we could put 100 unique marbles into a bucket and immediately it feels like the probability has an even distribution again. So it seems silly that the state of the other 99 marbles (whether they're all green or whether they're each unique) should have any effect on the probability of the red marble being chosen.

 

[Xarog]

I had in mind a simple system with parts (marbles) that aren't weighted or loaded in any way. For the sake of simplicity, would you agree with both these statements:

1.) In a bucket of 100 green marbles - each has a 1:100 chance of being selected.
2.) In a bucket of 99 green marbles and one red marble - each has a 1:100 chance of being selected.

No matter how big the bucket of marbles becomes, the probability of the unique red marble being selected will have the same probability as any other marble being selected. Again - it feels like it should be more improbable that the red marble be chosen, but we could put 100 unique marbles into a bucket and immediately it feels like the probability has an even distribution again. So it seems silly that the state of the other 99 marbles (whether they're all green or whether they're each unique) should have any effect on the probability of the red marble being chosen.

Yes, I would agree with those two statements (sort of).

In reality, a bucket has dimensions and you know the marbles lying near the top are more likely to be selected simply because you have to get past the marbles at the top to get to the marbles at the bottom. Etc.

All I was trying to say is that statistics always misrepresents reality in some form through this oversimplification. Thus it can never be objective no matter what, and no such formal expression as the one you were asking for exists, except for arbitrarily contrived scenarios where you can literally decide which variables will be at play, and to what degree.

 

[cguy]

If you assume that selecting each marble is equally likely (I.e., a "uniform distribution"), then the probability of picking any specific marble out of N marbles is 1/N. Thus if you have 1 red marble, the probability of choosing it is 1/N.

Soar, I think where you are getting confused is that while the probability of selecting any given marble is always 1/N, the probability of selecting a given class of marble depends on the distribution of that class, which is not uniform. So with N marbles, all green except one, there is a 1/N probability of the marble being red, and (N-1)/N probability of the marble being green. Note that as N increases (N-1)/N becomes closer to 1 (1 would be "certainty"), and the probability of choosing red ( 1/N ), goes to 0 ("certainly not").

So yes, there is a formal expression of this, but it requires that one assumes a uniform probability of any given marble being selected. Although this isn't strictly true in practice, with a good shuffle and a blind fold, any bias would be too minuscule to consider.

 

[saor]

Soar, I think where you are getting confused is that while the probability of selecting any given marble is always 1/N, the probability of selecting a given class of marble depends on the distribution of that class, which is not uniform. So with N marbles, all green except one, there is a 1/N probability of the marble being red, and (N-1)/N probability of the marble being green. Note that as N increases (N-1)/N becomes closer to 1 (1 would be "certainty"), and the probability of choosing red ( 1/N ), goes to 0 ("certainly not").

Thanks, that was helpful. Just to be clear that even if the probabilities are as follows:

Single red marble: 1/N
Entire green set: (N-1)/N

The probability for any particular green marble is also 1/N.

Kinda off-topic but a random number generator that uses atmospheric noise: https://www.random.org/
Some interesting posts there too...
https://www.random.org/analysis/#visual
https://www.random.org/randomness/
etc.

 

[dunkyd]

Question is, do you peep when leaning over the bucket ?