Difference between Pr(A | B) and Pr(A ∩ B)?

Question

Suppose A is a group of people who uses motorbike and B is people who uses car , and A ∩ B is group of people who uses both motorbike and car.

What is the probability of a person uses car , given that the person uses motorbike ?

By definition

Pr(A|B) ::= Pr(A∩B)/Pr(B)

If A ∩ B is group of people who uses both car and motorbike , why is the answer to question is Pr(A|B) and not the Pr(A∩B) itself ?

I'm sorry if this looks like a newbie question , but i am very confused and with full honesty don't know why. Nobody explained it to me.

Answer 1

The notation $P(A|B)$ represents a conditional probability, which is the probability of $A$ given that $B$ has occurred - it represents known or assumed prior information. By comparison, $P(A \cap B)$ is the intersection of the two events, so it's the probability that both $A$ and $B$ occur. I sometimes like to say that $|B$ means "in a universe where B is known to be true".

To give you an idea of why they're different things, consider what happens when $A$ is the event "I flip a coin and get heads" and $B$ is "I win the lottery". Then $P(A | B)$ means "in a universe where I'm holding a winning lottery ticket, what's the probability that when I flip a coin I get heads", whereas $P(A \cap B)$ means "what's the probability that I flip a coin and get heads ... and also win the lottery". Hopefully you can see that the former probability is much higher than the latter.

Answer 2

Think of the following example: In a given country there are $100$ citizens. $60$ of them just use the car, $15$ just use the motorbike and $5$ use both. We have $P(A)=.65$, $P(B)=.20$, $P(A\cap B)=.05$ and $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{0.05}{0.20}=.25.$$

Answer 3

Here are many better explanations about mathematical perspectives, but I'd like to explain more practical perspective.

In terms of $P(A|B)$, '|' says 'given, or fixed'. So $P(A|B)$ means probability of A when B has'already occured'.

Whereas $P(A \cap B)$ means that the probability of A and B occur in one time, not knowing any prior(old time) information.

Hence, when I knew that event B has occured, and wanting prob of A, should use $P(A|B)$, 'conditional probability', as the name means. On the other hand, when I don't know any information about B, and wanting prob of both A and B in the same time, ought to use $P(A \cap B)$.

This both of them make so much difference of'em.