Find the probability that you will talk with the number you dial for free. A faulty public phone is such that it returns the coin with probability $60\%$, gives you the number you dial with probability $20\%$, and it takes your coin but does not give you the required connection with probability $30\%$. Find the probability that you will talk with the number you dial for free. I get the answer $10\%$. It it correct?

$$P\\{\text{wrong number }\cup \text{ not free}\\}$$ $$=P\\{\text{wrong number}\\}+P\\{\text{not free}\\}-P\\{\text{wrong number }\cap \text{ not free}\\}=0.9$$

and

$$P\\{\text{correct number }\cap \text{ free}\\}=0.1$$