Probability and Random Experiments

[whitejac]

Homework Statement

Problem

Consider a random experiment with a sample space

S={1,2,3,⋯}.

Suppose that we know:

P(k) = P({k}) = c/(3^k) , for k=1,2,⋯,

where c is a constant number.

1. Find c.
2. Find P({2,4,6}).
3. Find P({3,4,5,⋯})

I am primarily interested in part 1, finding C. The rest should follow.

2. Homework Equations
I do not know of any relevant equations other than the three axioms of Probability:

For any even A, P(A) ≥ 0.
Prbability of the sample space S is P(S) = 1.
If a1, a2, a3 are disjoint events, then P(a1∪a2∪a3∪...) = P(a1) + P(a2) + P(a3)...

and the Inclusion Exclusion principle

The Attempt at a Solution

So, I'm a little confused on how to "find" C... I could start plugging in values for k, but then I would just be left with the limit as k → ∞ which would drop C to zero.
P(k= 1) = c/3
P(k=2) = c/9
P(k=3) = c/27
.
.
.
Since C is a constant, it seems a little bit trivial to find it as well as it will dramatically lose its impact in just a few values of k.

Also

My book uses a large Cup (∪) in some of its notations,
P(A) = P(large cup {s_j) = ∑P(sj)
(Beneath both the cup and the sigma are: sj ∈ A)
Does this mean the sum of the unions? I do not believe this helps the current question, but it is located in the vicinity of Random Experiments in my book.

 

[Ray Vickson]

Homework Statement

Problem

Consider a random experiment with a sample space

S={1,2,3,⋯}.

Suppose that we know:

P(k) = P({k}) = c/(3^k) , for k=1,2,⋯,

where c is a constant number.

1. Find c.
2. Find P({2,4,6}).
3. Find P({3,4,5,⋯})

I am primarily interested in part 1, finding C. The rest should follow.

2. Homework Equations
I do not know of any relevant equations other than the three axioms of Probability:

For any even A, P(A) ≥ 0.
Prbability of the sample space S is P(S) = 1.
If a1, a2, a3 are disjoint events, then P(a1∪a2∪a3∪...) = P(a1) + P(a2) + P(a3)...

and the Inclusion Exclusion principle

The Attempt at a Solution

So, I'm a little confused on how to "find" C... I could start plugging in values for k, but then I would just be left with the limit as k → ∞ which would drop C to zero.
P(k= 1) = c/3
P(k=2) = c/9
P(k=3) = c/27
.
.
.
Since C is a constant, it seems a little bit trivial to find it as well as it will dramatically lose its impact in just a few values of k.

Also

My book uses a large Cup (∪) in some of its notations,
P(A) = P(large cup {s_j) = ∑P(sj)
(Beneath both the cup and the sigma are: sj ∈ A)
Does this mean the sum of the unions? I do not believe this helps the current question, but it is located in the vicinity of Random Experiments in my book.

What is preventing you from using the "law" (for disjoint events)
$$P(A_1 \cup A_2 \cup A_3 \cup \cdots) = P(A_1) +P(A_2) + P(A_3) + \cdots ?$$
You know what are the probabilities $P(A_j) = P(k = j), j = 1,2,3, \ldots$, you know these events $A_j$ are disjoint and you should know what the event $A_1 \cup A_2 \cup A_3 \cup \cdots$ on the left represents, and so should know its probability.

 

[whitejac]

I think I understand that one. What I'm confused about, I guess, is understanding how to correlate that information into a value for C, unless C is the sample space which would be the limit as k increases in value.
(since for finite values, this number would be 1 but in the case of all positive integers then I'm guessing the sum of the unions would be the sample space as all of them would tally to that number?)

 

[Ray Vickson]

I think I understand that one. What I'm confused about, I guess, is understanding how to correlate that information into a value for C, unless C is the sample space which would be the limit as k increases in value.
(since for finite values, this number would be 1 but in the case of all positive integers then I'm guessing the sum of the unions would be the sample space as all of them would tally to that number?)

The above statements about C are nonsense; you yourself said in the OP that "c is a constant number". OK, so you don't happen to know yet what is the value of c---in fact, finding it is most of the problem. However, it is not random, does not change, and does not depend on any events; it is just some unknown input parameter. Basically, that is what you told us!

 

[whitejac]

Okay, I'm sorry. I have zero concept of this particular section. In my opinion the book and professor didn't really cover it.

So, I'm gathering that C is not a random variable. It can't be if it's a constant. That would be any value K.
So this would just be an arbitrary constant? That's what I gather from "basically this is what you told us!" But i don't see how to find it then.

 

[Ray Vickson]

Okay, I'm sorry. I have zero concept of this particular section. In my opinion the book and professor didn't really cover it.

So, I'm gathering that C is not a random variable. It can't be if it's a constant. That would be any value K.
So this would just be an arbitrary constant? That's what I gather from "basically this is what you told us!" But i don't see how to find it then.

Your problem description did not use the terminology "arbitrary", and neither did I. I said it was unknown. It is not ANY value K; it is some as-yet-undetermined value. (However, maybe we are not using language in the same way.)

If you use ALL the facts/assumptions underlying probability theory, you ought to see what missing bits of information you need to supply. I am not even sure your difficulties are with the probability aspects; you seem to be having trouble with some standard algebra. You are not using everything you have been taught in the past.

Aside from hinting to you to go back and re-read the fundamental "axioms" of probability, and maybe review basic algebra, I am not allowed to help you more without violating PF rules.

 

[Jonobro]

OP did you ever find C? I don't understand this problem...

 

[whitejac]

I did. I felt very foolish once I did. It has little do with probability and statistics and much more with ensuring your summation algebra makes sense. Review series, convergence, and use that information to find a way to convert the sum to 1.

 

[HallsofIvy]

Yes, c must be such that $\frac{c}{3}+ \frac{c}{3^2}+ \frac{c}{3^3}+ \cdot\cdot\cdot= \frac{c}{3}\left(1+ \frac{1}{3}+ \left(\frac{1}{3}\right)^2+ \left(\frac{1}{3}\right)^3+ \cdot\cdot\cdot\right)= 1$

"Geometric series".