I have to calculate
$$P(\text{Cavity} \mid \neg \text{tooth} \vee \neg \text{catch})$$
I have a table with probabilties
Now i've done it this way
$$\frac{P(\text{Cavity}, \neg \text{tooth} \vee \neg \text{catch})}{P(\neg \text{tooth} \vee \neg \text{catch})}$$
I've then done my calculations this way
$$P(\text {Cavity}, \neg \text{tooth} \vee \neg \text{catch})=P(\text{Cavity}, \neg \text{tooth})+P(\text{Cavity}, \neg \text{catch}) $$ And $$P(\neg \text{tooth} \vee \neg \text{catch}) = P(\neg \text{tooth})+P(\neg \text{catch})$$
Is this the right way? I've probably done it wrong. Can someone give me an advice? Thanks