Let a 52 deck of numbered cards be created as follows:

2 special cards: 0 and 1
25 powers of 2: 2, 4, 8, ..., 2^25
25 powers of 3: 3, 9, 27, ..., 3^25

Shuffle the deck and draw at random 3 cards. Evaluate the product of the 3 numbers, say P.

What is the probability of P=0?
What is the probability of P being a non zero integer square?
What is the probability of P being a 4-digit number?

  0 1 2 3 4 5 6 7 8 9 10111213
0                     x x x x
1                   x x x
2               x x x x
3             x x x
4         x x x
5       x x x
6   x x x
7 x x x
8 x
The matrix above shows the powers of 2 and powers of 3 
that lead to a 4 digit product.
Case 1: the 1 is drawn
subcase a: the other cards are both powers of 2: 4+5+5+6=20 ways
subcase b: one power of 2, one power of 3: 21 ways
subcase c: both powers of 3: 3 ways
Total case 1 = 47

Case 2: the 1 is not drawn
subcase a: all powers of 2: 4+5+7+8=24 ways
subcase b: two powers of 2, one power of 3: 13+14+8+5+4+1=45
subcase c: one power of 2, two powers of 3: 5+5+4+3+3+2+1+1=24
subcase d: all powers of 3: 3 ways 
Total case 2 = 96

For a grand total of 143
143/22100 = 11/1700 = 0.00647