Manufacturer produces TVs probability of a malfunction in TV is $p$. There are 3 different quality checks in the factory with probability of finding malfunction equal to $p_i$ for $i=1,2,3$ every TV goes only through one quality check and it's assigned to it at random. Later at the retailer warehouse there is another quality check with the probability of finding malfunction equal to $p_0$.
i) find the probability that TV with malfunction will get to the store? (i.e. goes through both checks)
ii) Assuming TV got to the store already what is the probability that it has a malfunction?
I described to events:
$A$- TV gets to the warehouse
$B$- TV lands in the store
Then: $$P(A\cap B)=P(B|A)P(A)=P(A|B)P(B)$$
but $P(A|B)$ is a probability that I goes through the first check if it goes through second check but the second check occurs only if it passes the first one so it must be equal $1$ but then $P(A\cap B) = P(B)$ what makes completely no sense. Later I thought that since these two events occur one after the other it's like throwing a dice twice so they are independent hence $P(A\cap B)=P(B)P(A)$ this makes more sense on the other hand it seems like if it passed the first test it should have bigger odds passing second one.
I feel like I have completely no intuition when it comes to this probability problems. Could some one clarify it for me please?
Edit: By probability of finding malfunction I mean probability of finding malfunction given that particular TV has one.