I have the following problem:
We suppose I have a company which builds houses. And I have a contract. I need to meet deadlines in end July, but that depends if I receive materials in middle July.
According to my experience, the probability of receiving materials middle of July are $\frac{2}{3}$ and $\pmb{if}$ I receive materials at time ( middle July), probability of meeting deadlines are $\frac{3}{4}$
So my question is : If I meet deadlines end of July , what would be probability that the cause were because I NOT received material at time (midde July).
If I model the problem I would have:
A = meets deadlines end July. So A = $\frac{3}{4}$
B= receive materials middle July. So B = $\frac{2}{3}$
I want to use a approach of Bayes theorem for this problem so:
Fist I calculate the probability that A $\pmb{and}$ B are met:
$P(A\bigcap B)= P(A\mid B)P(B)$
$P(A\bigcap B)= \frac{3}{4} * \frac{2}{3}$
$P(A\bigcap B)= \frac{1}{2}$
If I want to calculate Bayes' theorem to see probability that I finish house end of July but NOT receive materials at time I do :
$$P(B\mid A) =\frac{P(A\mid B)P(B)}{P(A\mid |B)P(B) + P(A\mid B^c)P(B^c)}$$
Problem is that I can't see $P(A\mid B^c)$ for calculate total probability.
Is it the same that $3/4$? but that doesn't make sense to me?