Probability of logical implication

Question

In my opinion, a carelessly formulated problem on my instructor's exam read:

$$P(\text{if the card is not a face card, it is divisible by 3})$$

It was graded graded as having the following meaning:

$$P(\text{not a face card $\land$ divisible by 3})$$

It seems to me that the vacuous case of the implication allows for face cards to be included in the event described by the implication. This would mean that the two formulations are not the same.

The course is not one in logic, does the instructor have grounds to argue that the two events are equivalent?

In general, can events be described in terms of logical implication, i.e., $P(p \Rightarrow q)$?

Answer 1

I see it like this:

event A = card is not a face card

event B = card is divisible by 3

the first probability you mention I would interpret like this:

P(B|A) = #cards_divisible_by_3 / (#all_cards - #face_cards) 
P(B|A) = 12 / (52 - 12) = 0.3 

the second probability you mention I would interpret like this

P(A,B) = #cards_divisible_by_3 / #all_cards
P(A,B) = 12 / 52 = 0.23

(all cards divisible by 3 are non-face cards)