Probability with an incomplete deck of cards?

"2 random cards are missing" is equivalent to "picked 2 cards randomly from the complete deck and discarded them". So let these discarded cards be X and Y.

And I assume that we picked 2 cards from the top of the incomplete deck and we found they are both 7s.

So we know that one of Event1, Event2, Event3 or Event4 occurred.

Event1. X,Y = 7,7 and 2 cards of 7 are picked
Its probability is (4/52)*(3/51)*(2/50)*(1/49) = 24 / (52*51*50*49)

Event2. X,Y = 7,not7 and 2 cards of 7 are picked
Its probability is (4/52)*(48/51)*(3/50)*(2/49) = 1152 / (52*51*50*49)

Event3. X,Y = not7,7 and 2 cards of 7 are picked
Its probability is (48/51)*(4/51)*(3/50)*(2/49) = 1152 / (52*51*50*49)

Event4. X,Y = not7,not7 and 2 cards of 7 are picked
Its probability is (48/51)*(47/51)*(4/50)*(3/49) = 27072 / (52*51*50*49)

Next, I assume that Event1 is contained in "the chance we are missing a 7 from the original deck". So the answer is

(24+1152+1152) / (24+1152+1152+27072) = 2328 / 29400 = 0.07918...