Ratios of conditional probabilities given a contingency table I'm having issues trying to understand what I need to use in order to get a correct answer for this homework problem. On top of that, I just really want to understand exactly what I should be doing for similar types of problems regarding conditional probabilities, because right now I'm lost.
I have two contingency tables, note that Gm stands for male, Gf stands for female, and Dn stands for Department N:
Table 1: Contingency table by gender and department (counting number of applicants)
       Da | Db | Dc | Dd | De | Df | Total
Gm   |825 |560 |325 |417 |191 |373 | 2691
-----|----|----|----|----|----|----|------
Gf   |108 |25  |593 |375 |393 |341 | 1835
-----|----|----|----|----|----|----|------
Total|933 |585 |918 |792 |584 |714 | 4526

Table 2: Contingency table by gender and department (counting the number of rejected applicants) 
       Da | Db | Dc | Dd | De | Df | Total
Gm   |313 |207 |205 |279 |138 |351 | 1493
-----|----|----|----|----|----|----|------
Gf   |19  |8   |391 |244 |299 |317 | 1278
-----|----|----|----|----|----|----|------
Total|332 |215 |596 |523 |437 |668 | 2771

QUESTIONS
a. Report the conditional probability that an applicant was rejected among male applicants, namely, $P(Rejected|G_M)$.
b. Report the conditional probability that an applicant was rejected among female applicants, namely, $P(Rejected|G_F)$.
c. From Table 1, report the six ratios of conditional probabilities:
$\frac{P(G_M|D_A)}{P(G_F|D_A)}...\frac{P(G_M|D_F)}{P(G_F|D_F)}$
d. Find the six ratios of conditional probabilities:
$\frac{P(Rejected | G_M \cap D_A)}{P(Rejected | G_F \cap D_A)}...\frac{P(Rejected | G_M \cap D_F)}{P(Rejected | G_F \cap D_F)}$
MY ANSWERS
a. $\frac{1493}{2691} = 0.555$. This makes sense to me logically, but not when using the formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$, how could I even apply this formula? How can I even calculate $P(Rejected \cap G_M)$?
b. Applying the same logic to part a, I can do $\frac{1278}{1835} = 0.697$ But once again how can I calculate this with the formula, as a way to prove it to myself?
c. I just want to know if I'm on the right track. For $\frac{P(G_M | D_A)}{P(G_F | D_A)}$ would I do $\frac{825/933}{108/933} = 7.639$?
d. How can I apply a formula to this? And also would I just do $\frac{313}{825} = 0.379$?
I guess you can see my main issue is proving to myself that I'm right with the formulas I've been given. If anyone could offer some advice on how to use these formulas, or what they are used for I would appreciate it.
 A: (a) Your answer is right. $P(\text{Rejected}\cap G_M)$ is the $G_M/\text{Total}$ cell in Table $2$ (with value $1493$) divided by the $\text{Total}/\text{Total}$ cell in Table $1$ (with value $4526$). It is the proportion of all $4526$ applicants that are both rejected and male (i.e. "$\text{Rejected}\cap G_M$"). So
$$P(\text{Rejected}\mid G_M) = \dfrac{P(\text{Rejected}\cap G_M)}{P(G_M)} = \dfrac{1493/4526}{2691/4526} = \dfrac{1493}{2691}.$$
(b) Similarly, $P(\text{Rejected}\cap G_F)$ is the $G_F/\text{Total}$ cell in Table $2$ (with value $1278$) divided by the $\text{Total}/\text{Total}$ cell in Table $1$ (with value $4526$). It is the proportion of all applicants that are both rejected and female (i.e. "$\text{Rejected}\cap G_F$"). Your answer is right and the method is similar to part (a).
(c) Your answer is right. The method is
$$\dfrac{P(G_M\mid D_A)}{P(G_F\mid D_A)} = \dfrac{P(G_M\cap D_A)/P(D_A)}{P(G_F\cap D_A)/P(D_A)} = \dfrac{P(G_M\cap D_A)}{P(G_F\cap D_A)} = \dfrac{825/4526}{108/4526} = \dfrac{825}{108}.$$
because, for example, $P(G_M\cap D_A)$ is the $G_M/D_A$ cell in Table $1$ (with value $825$) divided by the $\text{Total}/\text{Total}$ cell in Table $1$ (with value $4526$).
(d) Here we have
$$\dfrac{P(\text{Rejected}\mid G_M\cap D_A)}{P(\text{Rejected}\mid G_F\cap D_A)} = \dfrac{P(\text{Rej.}\cap G_M\cap D_A)/P(G_M\cap D_A)}{P(\text{Rej.}\cap G_F\cap D_A)/P(G_F\cap D_A)} = \dfrac{\frac{313}{4526}/\frac{825}{4526}}{\frac{19}{4526}/\frac{108}{4526}} = \dfrac{313/825}{19/108}.$$
because, for example, $P(\text{Rej.}\cap G_M\cap D_A)$ is the $G_M/D_A$ cell in Table $2$ (with value $313$) divided by the $\text{Total}/\text{Total}$ cell in Table $1$ (with value $4526$). It is the proportion of all applicants that are rejected and male and in department A (i.e. "$\text{Rejected}\cap G_M\cap D_A$").
