Can't find intersection of two probabilities.

Question

I have the following problem:

While producing goods, defect through event A has 3% probability and defect through event B has 4% probability. Total goods that are not defected - 95%. Find correlation coefficient between A and B.

My steps are:

$\rho$ = $\text{Cov}(A,B)\over\sqrt{\text{Var}(A)\text{Var}(B)}$

$\text{Cov}(A,B) = E[AB] - E[A]E[B] = P(A=1,B=1) - P(A)P(B)$

$\text{Var}(A) = p(1-p)$ as it is Bernoulli.

I'm having problem at finding $P(A=1,B=1)$:

$P(A=1,B=1)$ doesn't equal to $P(A) + P(B) - P(A)*P(B)$ because as I understand

they are dependent and $P(A\cap B) \neq P(A)*P(B)$

So my question is how to find the quantity $P(A \cap B)$ in my case.

I guess there is something with the fact that 95% total goods are not defected, but don't know how to use it.

Thanks!

Answer 1

Use the fact that $\mathbb P[(A \cup B)^c] + \mathbb P[A] + \mathbb P[B] - \mathbb P[A \cap B] = 1$.

In particular, $$\mathbb P[A \cap B] = 0.95+0.03+0.04-1= 0.02$$ so you can plug this into your method and you are done.

Answer 2

$P(A\cap B)=P(A\cup B)-P(A)-P(B)$ then you can the fact that 95% of the goods are not deflected.