Is it right to condition a RV on dependent RV?

[devonho]

Homework Statement

Let there be two discrete random variables:

[itex]
X \in \lbrace 1,2,3,4,5,6,7,8,9,10 \rbrace \quad \text{where } P[X] \text{ is uniformly distributed over the sample space of } X \text{.}
[/itex]

[itex]
B = \left\lbrace
\begin{array}{cl}
1 & \text{if} \quad X>4 \\
0 & \text{otherwise}\\
\end{array}\right.
[/itex]

[itex]
P[B \mid X]=\left\lbrace
\begin{array}{cl}
0 & \text{if} \quad x \in \lbrace 1,2,3,4 \rbrace\\
1 & \text{if} \quad x \in \lbrace 5,6,7,8,9,10 \rbrace\\
\end{array}\right.
[/itex]

[itex]
P[X] = {1\over10}
[/itex]

[itex]
P[X,B] = P[B \mid X]P[X] = \left\lbrace
\begin{array}{cl}
0 &\text{if} \quad x \in \lbrace 1,2,3,4 \rbrace\\
{1\over 10} & \text{if} \quad x \in \lbrace 5,6,7,8,9,10 \rbrace\\
\end{array}\right.
[/itex]
The above should be agreeable. But what about:

[itex]
P[X,B] = P[X \mid B]P
[/itex]

Since B is dependent on X, is it meaningful or even correct to write an expression for P[X|B]?

Homework Equations

The Attempt at a Solution

I think no because the conditional probability will then be recursive.

 

[haruspex]

It's perfectly ok to discuss the probability of an underlying event given an observation concerning it.
P[B=0] = 0.4
P[X=1|B=0] = 1/4
P[X=1 & B=0] = 0.1
etc.