Probability of "2,3 or 4 Machines Not in Use

[riemann86]

Hello

I have not taken a subject in set theory, only in statistics. Maybe you guys can help me, I want to describe one of the events with the other, and I am wondering if one can do it.

Lets say that we have 6 machines, of these 0, 1,2,3,4,5 or 6 of them can be in use. That is, we distingiuish how many are in use, not which particular one we are using.

Now let's say that you want to find the probability that "not 2 or 3 or 4 machines are in use".
If we say that the event A is {2 machines, or 3 machines or 4 machines in use}
then we want to find P(not A) = 1-P(A), this is easy.

Now is the tricky part, look at the event:
"2,3 or 4 machines are not in use". First I thought that this was the same as the first one, but it is actually the same as the event A, because if 2 is not in use, then 4 is in use, and if 3 is not in use then 3 is in use, and if 4 is not in use then 2 is in use.

So we have that "2,3 or 4 in use" = "2,3,4 not in use" and this does not equal " not 2,3,4 in use"

My question is if this can be shown with something deeper, than just going over all the different possibilities? For instance, could we say that the event "2,3 or 4 machines are not in use" is "not not A" and hence it becomes A? Or is it another way to describe "2,3 or 4 machines are not in use" in terms of A, and then reduce this expression so you get to A?

 

[Stephen Tashi]

Hello
Now let's say that you want to find the probability that "not 2 or 3 or 4 machines are in use".

English is ambiguous. It isn't clear whether that phrase means: "(not 2) or 3 or 4" or whether it means "not (2 or 3 or 4)" - i.e. "neither 2 nor 3 nor 4". I assume you're interpreting it as the latter.

Now is the tricky part, look at the event:
"2,3 or 4 machines are not in use". First I thought that this was the same as the first one, but it is actually the same as the event A

What you mean by "the first one" and what do you mean by "event A"?

There is another problem with English in the phrase "2,3 or 4 machines are not in use". In common speech the statement "2 machines are not in use" means exactly 2 machines are not in use. In mathematics, the technical interpretation of "2 machines are not in use" is "at least 2 machines are not in use". Unfortunately, homework problems in math books sometimes use common English and sometimes use mathematical English.

because if 2 is not in use, then 4 is in use, and if 3 is not in use then 3 is in use, and if 4 is not in use then 2 is in use.

That's a correct deduction if "2 not in use" means "exactly 2 are not in use".

My question is if this can be shown with something deeper, than just going over all the different possibilities? For instance, could we say that the event "2,3 or 4 machines are not in use" is "not not A" and hence it becomes A? Or is it another way to describe "2,3 or 4 machines are not in use" in terms of A, and then reduce this expression so you get to A?

I don't think you can do the deduction from simple set theory on a set that has 7 elements. Suppose the set is {a,b,c,d,e,f,g}. If E is an event then you cannot deduce from basic set theory that d \not \in E implies e \in E.

 

[riemann86]

English is ambiguous. It isn't clear whether that phrase means: "(not 2) or 3 or 4" or whether it means "not (2 or 3 or 4)" - i.e. "neither 2 nor 3 nor 4". I assume you're interpreting it as the latter.

I mean the latter yes.


What you mean by "the first one" and what do you mean by "event A"?

With the first one I meant: "neither 2 nor 3 nor 4"
The event A is the event I defined over I labeled it A, it is "2 or 3 or 4 in use".

There is another problem with English in the phrase "2,3 or 4 machines are not in use". In common speech the statement "2 machines are not in use" means exactly 2 machines are not in use. In mathematics, the technical interpretation of "2 machines are not in use" is "at least 2 machines are not in use". Unfortunately, homework problems in math books sometimes use common English and sometimes use mathematical English.

With "2,3 or 4 machines are not in use" I mean exactly 2 or exactly 3, or exactly 4, not in use.

.....

 

[riemann86]

I don't think you can do the deduction from simple set theory on a set that has 7 elements. Suppose the set is {a,b,c,d,e,f,g}. If E is an event then you cannot deduce from basic set theory that d \not \in E implies e \in E.

Ok, so there isn't a similar way to describe :

"(exactly 2 or exactly 3 or exactly 4) is not in use"
in terms of "exactly 2, or exactly 3 or exactly 4 is in use"

Like we can describe: "not( exactly 2, or exactly 3 or exactly 4) in use" is the complement of "exactly 2 or exactly 3 or exactly 4 in use"?

But we can say that if a set V contains numbers from 0 to N, that symbolises an exact amount of units active. And another set V2 contains numbers from 0 to N that symbolises an exact amount out of of N units inactive. Then V = V2 if and only if one can find the numbers are so that v = N-v2, for every number v and v2 in V and V2. And it is just a coincidence that this happened in this example?

 

[riemann86]

Maybe it will be clearer what I am struggling with if I use one number instead of three.

Let's redefine the event A to "exactly 1 machine in use".
Then let's define the event B = "not (exactly 1 machine in use)", is well defined as the complement of A.
So if the sample space is S, we have B = S-A.

If we define the event C = "exactly one machine is not in use". We have that 5 machines must be in use.
But can we write C = function of A, the way we wrote B = S-A?

 

[Stephen Tashi]

But can we write C = function of A, the way we wrote B = S-A?

As I said before, on a set S of 7 arbitrary elements, I don't think you can write any expression using set operations that accomplishes what you want because what you want is not a property of an arbitrary set of 7 elements.

Your set S is {u0, u1,u2,...u6}. The phrase "exactly 2 machines are in use" is simply a name for the element u2. Unless you specify a meaning, the phrase "exactly 2 machines are in not in use" doesn't refer to any particular element of S or subset of S. For example, suppose we had specified that the name of "u2" is "Pink is beautiful". This does not tell us what element of S or what subset of S would be represented by the phrase "not pink is beautiful" or "pink is not beautiful" or "purple is beautiful".