How do you get joint probability of two events conditional on the same event?

Question

I want to get the joint probability of two independent events, both conditional on another event.

What I know:

Probability of A is 0.2375.

Probability of B given A is .8.

Probability of C given A is .5.

Now, I know that B and C is a joint probability (B * C). I'm also fairly certain that the "un-conditional" probabilities of B and C separately (let's call them Bu and Cu) are (A * B) and (A * C). My question is, is the joint probability of B and C calculated as (A * Bu * Cu), or is it ((A * Bu) * (A * Cu))?

Answer 1

Are you told that events $B$ and $C$ are "independent", or are they "conditionally independent given $A$"?

• If $B$ and $C$ are independent, then, by definition: $\mathsf P(B, C)=\mathsf P(B)\,\mathsf P(C)$. However you do not appear to have enough information to find this, since you cannot use the Law of Total Probability without the conditional probability of $B$ given the complement of $A$, nor without that for $C$.

$$\mathsf P(B)=\mathsf P(B\mid A)\,\mathsf P(A)+\mathsf P(B\mid A^\complement)\,(1-\mathsf P(A))\\\mathsf P(C)=\mathsf P(C\mid A)\,\mathsf P(A)+\mathsf P(C\mid A^\complement)\,(1-\mathsf P(A))$$

• On the other hand, if $B$ and $C$ are conditionally independent given $A$, then you can at least find their joint conditional probability given $A$. This is $\mathsf P(B, C\mid A)=\mathsf P(B\mid A)\,\mathsf P(C\mid A)$. However you still cannot find their joint probability unconditionally for the same reason.

$$\mathsf P(B,C)=\mathsf P(B,C\mid A)\,\mathsf P(A)+\mathsf P(B,C\mid A^\complement)\,(1-\mathsf P(A))$$