What is the intuition behind the probability intersection formula?

Question

I am trying to learn the maths I need for data science, but I left formal maths education so long ago that my maths intuition is incredibly weak. I am confronted with the following explanation, and I just don't see it as obvious in the way that the author expects me to:

"If A and B are not independent, the equation (1.6) generalizes to

 P(A and B) = P(A) P(B|A) 

This should make sense to you. Suppose 30% of all UC Davis students are in engineering, and 20% of all engineering majors are female. That would imply that 0.30 x 0.20 = 0.06, i.e., 6% of all UCD students are female engineers."

Why are we multiplying P(A) and P(B|A)? Of course, I know that

 𝑃(𝐴∣𝐵)=𝑃(𝐴∩𝐵)/𝑃(𝐵)

and so rearranging we get the above formula, but that doesn't help explain why (for me). I've worked through his explanation of the female engineering students assuming that the overall number of students is 100, but I'm not seeing how this connects to the formula.

Clarification: equation (1.6) is:

P(A and B) = P(A) · P(B)

when A and B are independent.

Answer 1

We get to multiply the probabilities of individual events in a compound event when the outcome of the second doesn't depend on the first. So if I want to choose between the numbers 1-3 on the first number and 1-5 on the second and repeats are allowed, I get to just multiply $\frac 1 3$ by $\frac 1 5$.

We can view $A \cap B$ sort of the same, except for that pesky "they aren't independent". However, we can force independence! For my first probability, I just need $A$ to be true. Now, we want to shrink our universe of possibilities to only those in which $A$ is true. From that pool, we then can look at the probability of $B$ happening "Next" as independent. But that probability is exactly $P(B|A)$.

Rather than the formula viewpoint of $P(B|A)$, I prefer the "restricted universe" viewpoint. $P(B)$ is the probability of $B$ happening in the big universe. $P(B|A)$ looks at what happens if we restrict our viewpoint to only the parts in which $A$ happened, then rerun the problem of the chances of $B$ happening in our sub-universe.

Answer 2

Probability can be defined as the size of the set of favourable outcomes over the size of the set of outcomes.

$$P(A)=\frac{|A|}{|\Omega|}$$

In your example the sets involved are:

• $\Omega$: The set of all students at UCD
• $A$: The set of engineers at UCD
• $B$: The set of females at UCD
• $A\cap B=A|B=B|A$: The set of female engineers at UCD

However the probabilities are different, depending on the sample space.

• $P(\Omega)=1$
• $P(A)=\frac{|A|}{|\Omega|}$
• $P(B)=\frac{|B|}{|\Omega|}$

as expected. But $A\cap B$ and $B|A$ have different sample spaces, namely $\Omega$ and $A$ respectively.

• $P(A\cap B)=\frac{|A\cap B|}{|\Omega|}$
• $P(B|A)=\frac{|A\cap B|}{|A|}$

and the formula becomes

$$\frac{|A\cap B|}{|\Omega|}=\frac{|A|}{|\Omega|}\frac{|A\cap B|}{|A|}$$