What is the size of A ∩ B? There is a 4 digit PIN.
I know that there are 10000 numbers are available.
Assume A stars with 0 => In this case, 1000 numbers are available. (0000~0999)
Assume B starts with 0 and ends with 0. 100 numbers are available.
Can you tell me they are correct or not? and answer 5 as well.

*

*Pr(A) is 0.1 (1000/10000 => A's availabilities/total availabilities)


*Pr(B) is 0.01 (100/10000 => B's availabilities/total availabilities)


*Pr(A ∩ B) is the same as Pr(B) so, 0.01


*Pr(B|A) is 0.1


*What is the size of A ∩ B is 100? or what should I have to write?
 A: I assume the random process you are considering is the following:
Randomly select a PIN number $x$ from the state space of all possible PINs $U$. Then $A$ is the event $x$ starts with 0, and $B$ is the event $x$ both starts and ends with 0.
Then we have $\vert U\vert=10000$, $\vert A\vert=1000$, and $\vert B\vert=100$
The answers to (1) and (2) follow from the sizes of the events and state space. $P(A)=\frac{\vert A\vert}{\vert U\vert}=\frac{1000}{10000}=0.1$.
Now for question (3) you note that the probability is the same as (2). This is because you know, on some intuitive level, that $A\cap B$ is just as likely as $B$. What might not be clear is that this is because of the answer to (5). See, the formula used for (3) is the same as the one used in (1) and (2).
$$P(B)=\frac{\vert B\vert}{\vert U\vert}=\frac{100}{10000}=0.01$$
$$P(A\cap B)=\frac{\vert A\cap B\vert}{\vert U\vert}=\frac{100}{10000}=0.01$$
Note that contained within this, we have the equality $\vert A\cap B\vert=100$

For a more broadly applicable solution, we need to first solve (5) and use the answer in (3). So we need to find the size of the event $A\cup B$.
Note that $B$ is a subset of $A$. This means that every outcome in $B$ is also in $A$. Now $A\cap B$ is the set of outcomes which are in both $A$ and $B$. But if an outcome is in $B$, it's in $A$ also, so $B=A\cap B$. Since the events are equal, it follows that their sizes are equal: $\vert A\cap B\vert=\vert B\vert=100$.
