The binary variables $A,B,C \in \{0,1\}$ are not independent. I need to solve for the probability of $A$ and $B$ occurring simultaneously given that $C$ and $A$ occur: $P[A=1, B=1| C=1, A=1]$.
I have expanded the probability using the Bayes theorem, but I'm not sure it's correct. $$ Lemma1: P[a=1, b=1|b=1] = P[a=1, b=1]\\ Lemma2: P[a=1,b=1|c=1] = P[d|c=1], d=\{a=1,b=1\}\\ Lemma3: P[d|c=1,a=1] = \frac{P[c=1|d,a=1]*P[d|a=1]}{P[c=1|a=1]}\\\ \\ P[A=1, B=1| C=1, A=1] = P[\alpha|C=1, A=1]; \alpha=\{A=1, B=1\}\\ P[\alpha|C=1, A=1] = \frac{P[C=1|\alpha,A=1]*P[\alpha|A=1]}{P[C=1|A=1]}\\ = \frac{P[C=1|A=1, B=1,B=1]*P[A=1, B=1|A=1]}{P[C=1|A=1]}\\ = \frac{P[C=1|A=1, B=1]*P[A=1, B=1]}{P[C=1|A=1]} $$ That's as far as I've come. Are my lemmas and process correct? Is there a way to further simplify the probability? If possible, I'd like to express the probability in terms of the quantities I have information for: $P[A=1], P[C=1], P[A=1|C=1], P[B=1,C=1]$
Note: Lemma 3 came from Bayes rule for two conditioned events