Can I calculate this probability? I need some help with a small probability question someone asked at work:
I have 3 random variables: $B, A, L$.
$A$ and $L$ are independent, and I need to calculate:
$$
P(B|A,L)
$$
But I only know the values of $P(B|A)$ and $P(B|L)$.
Can I do this?
Thanks!
 A: No, you do not have enough information.
For example, consider looking at the calendar and the window when you wake up on a randomly chosen day.  Let $A$ represent the day of the week, and $L$ represent whether it is raining or not.  These are clearly independent in general.
Now suppose $B$ is the event that you will play basketball today.  Suppose you play on Tuesdays at the park when it is not raining, and on Thursdays if it's raining then you play basketball in the gym instead of going running.  Neither $P(B|A)$ nor $P(B|L)$ can tell you this information.  But this information is contained in $P(B|A,L)$.  So you cannot calculate $P(B|A,L)$ from just $P(B|A)$ and $P(B|L)$.
Mathematically, if you write the values of $P(B|A,L)$ as a matrix with rows for values of $A$ and columns for values of $L$, then $P(B|A)$ gives you the sum of each row, and $P(B|L)$ gives you the sum of each column.  This is not enough information to give you all the matrix entries.
The fact that $A$ and $L$ are independent is irrelevant if $A$ and $L$ are given, as in $P(B|A,L)$.
