Is P(A and B) = P(B and A)? I was wondering if the following statement is true and if there are any times in which the following is not true:
Is the probability of A and B equal to the probability of B and A?
P(A and B) = P(B and A)?
Would it be suffice to show that they are equal based on P(A and B) representing the same intersection as P(B and A) on a venn diagram?
 A: The probability of events $A$ and $B$ both occurring is the same as the probability of $B$ and $A$ both occurring. I know it seems almost silly and obvious when written like that.
This is denoted by $p(A\cap B) = p(B \cap A)$.
Note that this is quite distinct from the following two scenarios:
The probability of $A$ occurring if $B$ occurs is not necessarily the same as the probability of $B$ occurring if $A$ occurs. This has to do with conditional probability and the two probabilities are denoted $p(A|B) $ and $p(B|A) $ respectively.
Also, when there is a temporal (time) or sequential aspect involved, note that the probability of $A$ occurring, then $B$ is not necessarily the same as when the situation is reversed.
Your main question seems to have been addressed adequately in the comments, but I wanted to add these caveats so you get a fuller picture. 
A: Yes.  
Events are sets of outcomes; that is why we use set operators for intersection and union to represent "and" and "or" connections between events.
The intersection of two sets is the same whichever order they sets are written. $$A\cap B=B\cap A$$
The probability measure for that intersection is thus the same.$$\mathsf P(A\cap B)=\mathsf P(B\cap A)$$

Naturally, the same argument applies for unions of events.$$\mathsf P(A\cup B)=\mathsf P(B\cup A)$$
