Probability of a car accident given that a man is driving (Bayes formula)

Question

I came across the following exercise illustrating the Bayes formula:

Let's consider the events: F = {female driver}, G = {male driver}, and A = {car accident}. We also know that P(A|F) = a, P(A|G) = b, and P(F) = P(G) = 1/2. Compute P(A).

Then the solution simply gives the formula for computing P(A). But I don't understand the meaning of events such as $A \cap F$. When we define the events F, G and A, what is the sample space $\Omega$? Is it the cartesian product of two sample spaces $ \Omega_1 = \{$female driver, male driver$\}$ and $ \Omega_2 = \{$accident, no accident$\}$?

In this case, the events should be defined as follows: F = $\{$ (female driver, accident), (female driver, no accident) $\}$, and A= $\{$ (female driver, accident), (male driver, accident) $\}$. Therefore, events such as $A \cap F$ can be properly defined: $A \cap F$ = $\{$ (female driver, accident) $\}$.

Is my understanding correct?

Answer 1

Hint -

In this question intersection of two events not given. You have to find that first.

For example -

$P(A|F) = \frac{P(A \cap F)}{P(F)}$