Why does P(A ∩ B|C) = P(A|C)P(B|C)?

Question

I understand that if A and B are disjoint, then P(A U B|C) = P(A|C) + P(B|C) and there is an analogous relationship with P(A ∩ B|C). However I can interpret the former intuitively via a Venn Diagram. But I don't understand the intuition behind P(A ∩ B|C) = P(A|C)P(B|C) . Thanks for the help.

Answer 1

The intuition behind $$P(A ∩ B|C) = P(A|C)P(B|C)$$ is simple. It basically states that $A$ and $B$ are independent given $C$. This is also called conditional independence.

Here is a great link

https://towardsdatascience.com/conditional-independence-the-backbone-of-bayesian-networks-85710f1b35b

$$P(A ∩ B|C) = P(A|C)P(B|C)$$ is also referred to as $$A ㅛ B | C$$

As you asked for intuition, here are some examples:

• Amount of speeding fine $ㅛ$ Type of car | Speed

why? Because given the speed of the car, the fine is independent of the type of the car. Note that if we do not condition on speed, the fine and the type of car are dependent, as some cars (such as toys) cannot reach high speeds.

• Lung cancer $ㅛ$ Yellow teeth | Smoking

why? If we do not know if the person smokes, cancer and yellow teeth are not independent. If you have one, you are more likely to have the other. Now, if you condition on smoking, i.e., if you take only those that smoke, then cancer and yellow teeth may be independent, as in this subset of people there is likely no relationship between the two events.

• Child’s genes $ㅛ$ Grandparents’ genes | Parents’ genes

why? Because the parent's isolate children from grandparents

Answer 2

As was mentioned above, this is true iff A and B are conditionally independent given C.

These are a further application of the addition rule and multiplication rule of elementary probability:

If $A,B$ are independent events then $$ P(A \cap B) = P(A) P(B)$$

If $A, B$ are disjoint, then

$$P(A\cup B) = P(A) + P(B)$$