Conditional Probability/Restrictions

I'm looking to get some insight into a moderately challenging conditional probability problem:

Consider a sequence of random variables , ..., which each take the values 0 and 1. Assume that

Pr( = 1) = 1 - Pr( = 0) = , j =1,...,n

where 0 < < 1 and that

Pr( = 1| =1) = , j = 2,...,n.

(a) Find Pr( = 0| =1), Pr( = 1| =0),Pr( = 0| =0).

(b) Find the requirements on so that this describes a valid probability distribution for , ..., .

So for part (a) I have: 1- , , and 1- , respectively.

I do not know how to begin part (b).

Re: Conditional Probability/Restrictions

For part (a), you may have some mistakes. Here is what I get:

implies

so I agree that .

But,

Since , we find

So,

Then

Let's check that we have everything correct:

So, based on this, we have

Re: Conditional Probability/Restrictions

Thanks for the help.

Did you have any idea on the restrictions for ?

Obviously the entire set has to be equal to a probability of 1. So are the bounds for are 0 < < 1?

Re: Conditional Probability/Restrictions