Given the probabilities $P(k=10), P(k=11), P(k=12), P(k=13)$ and the probabilities $P(a= 0, b= 0|k)$, $P(a= 0, b= 1|k)$, $P(a= 1, b= 0|k)$, $P(a= 1, b= 1|k)$ for all 4 values of $k$. What is $P(a|k=10)$?
I found a formula stating:
$$P(a|k) = \frac{P(a,k)}{P(k)} $$
I know $P(k=10)$ but I don’t know $P(a,k)$. Is there a way to find $P(a,k)$ from $P(a,b|k)$ or is there another way to find $P(a|k)$ from the given data?
The expression $P(a|k=10)$ means the probability of $a$ being $1$ given that the event $k=10$ has already occured. This makes the probability $P(k=10)$ redundant. The asked probability can hence be computed as:
$P(a|k=10) = P(a=1|k=10) = P(a=1,b=0|k=10) + P(a=1,b=1|k=10)$