What is the intuition behind probability theorem? If $A\subset B$ then $P(B|A) = 1$.

Question

If $A\subset B$ then $P(B|A) = 1$.

I don't quite understand this. Let's say $A = (4,3)$ is the result of a roll of two dice. $A\subset B$ where $B$ is the set of all ordered pairs of dice rolls. But the fact that $A$ happens doesn't imply that every dice roll in $B$ happens.

Can someone explain why $B$ in this scenario is guaranteed to happen?

Answer 1

Just think about this Venn diagram:

If $A$ happens, then it is certain that a $B$ happens... but it doesn't matter that not every $B$ happens.

Answer 2

Definition: event $A$ happen if the outcome of experiment,$\omega$ , be in $A$

let $\omega$ is the result of experiment, so if $A$ happen this means $\omega\in A$ so

$\omega\in B$ so $B$ happened.

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

Answer 3

You can read $A \subset B$ as "if $A$ happens, then $B$ happens". Now $P(B|A)=1$ is obvious.

And to clarify, for an event to happen does not mean every possible sub-event has occurred. It simply means one such sub-event occurred.

Answer 4

As I see your query, I think you mean that it is not B which has occured as it would need more sample points. And thats right, but here we are calculating P(B/A) i.e. probability of occurance of B if A has already occured. so we are not calculating probability of B but a conditional probability when we have limited our sample space to A and from within A we wish to calculate the probability of occurance of B