Geometric interpretation of multiplication of probabilities? When dealing with abstract probability space $\Omega$ which consists of atomic events with measure ($P: \Omega \rightarrow \mathbb{R}$) defined for them, it seems natural to start immediately imagining simple cases like this: $\Omega$ is some closed area in 2-D space partitioned into subareas, representing atomic events. Measure is the area. The total area of $\Omega$ is 1.
In fact this is what they sometimes picture in textbooks with the help of vienne diagrams or whatever.
This intuition works fine for simple cases, especially with adding probabilities.
But then it comes to probability multiplication: I don't understand how to interpret it within this simple model. Is there a way?.. Is there at all some mental model to think about probability multiplication in simple geometric terms, besides Lebesgue terms?
Thank you!
 A: The point of this answer is to help develop a geometrical intuition of probability; for this reason, I sacrificed precision to simplicity.
Explanation: Part 1
We can view probabilities as fractions of populations. For example, about half the population is female, so the probability for an equiprobably chosen person to be female is approximately $\frac{1}{2}$. We can view areas as populations of points.
To multiply an area by a fraction is to take a corresponding subarea, and to divide an area $A$ by an area $B$ is to determine how much bigger $A$ is than $B$. For example, a square $S$ can be divided into two rectangles $R$ and $R'$ of equal area; this area is half the area of $S$, that is to say: $\frac{\operatorname{area}(R)}{\operatorname{area}(S)} = \frac{1}{2}$ or $\frac{\operatorname{area}(S)}{\operatorname{area}(R)} = 2$ or $\operatorname{area}(R) = \frac{1}{2} \times \operatorname{area}(S)$. The ability to geometrically visualize these identities should be very helpful in what follows.
Conventions
Let $\Omega$ be a square of area $1$. Let $A$ and $B$ be overlapping rectangles within $\Omega$. Let $C := A \cap B$ be the rectangle where A and B overlap. By all means, draw a colored illustration, I found it helpful.
I will use the same notation $T$ for a given rectangle $T$, its area “$\operatorname{area}(T)$”, and the event “$x \in T$”, because it helps intuition to identify these things. I will thus identify a shape’s area with its set of points.
Explanation: Part 2
First, we have to visualize P(A|B), the probability that $x \in A$ assuming that $x \in B$.
\begin{align}
P(A|B) & = \text{the fraction of points of B which are in A} \\
& = \text{the fraction of points of B which are in C} \\
& = \frac{C}{B} \\
& = \frac{P(A \cap B)}{P(B)}.
\end{align}
I will now proceed to build an intuition of Bayes' theorem. We see that $P(A|B) = \frac{C}{B}$. A way to calculate $\frac{C}{B}$ is to first approximate it by $\frac{A \times d}{B}$, where $d$ is a corrective constant. Instead of working within B, we now work within A: we want to select a fraction $d$ of A. The fraction of A that we want is $\frac{C}{A}$, so $d = \frac{C}{A} = P(C|A)$. This allows visualizing Bayes’ theorem:
$P(A|B) = \frac{C}{B} = \frac{A}{B} \times \frac{C}{A} = \frac{P(A)}{P(B)} \times P(C|A) = \frac{P(A)}{P(B)} \times P(B|A)$.
