Random experiment and sample space

[Jonobro]

Homework Statement

[/B]
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,⋯})

Homework Equations

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)...

The Attempt at a Solution

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If I plug in values for k, as k increases the probability will decrease.
P(k= 1) = c/3
P(k=2) = c/9
P(k=3) = c/27
However, I am not understanding two main things.
How am I supposed to know when k stops increasing, or does it go to infinity.
How am I supposed to find the c value without being given any other information?
Any help is appreciated.

 

[andrewkirk]

You need to calculate $\sum_{k=1}^\infty (\frac{1}{3})^k$.
If you are being given this problem then you should have been taught how to find out the limit of an infinite geometric series, which is a standard technique in introductory calculus. Check your textbook, or perhaps the textbook for whatever maths courses were prior to this one. Or else just read the wikipedia article on geometric series, which explains how to do it.

 

[Ray Vickson]

Homework Statement

[/B]
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,⋯})

Homework Equations

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)...

The Attempt at a Solution

[/B]
If I plug in values for k, as k increases the probability will decrease.
P(k= 1) = c/3
P(k=2) = c/9
P(k=3) = c/27
However, I am not understanding two main things.
How am I supposed to know when k stops increasing, or does it go to infinity.
How am I supposed to find the c value without being given any other information?
Any help is appreciated.

The value k never "stops". Do the integers (whole numbers) ever stop? Well, in this case the "sample space" consists of all the whole numbers $\{ 1,2,3,\ldots \}$.