14
17
15
16

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We know that 44 and 38 are even numbers and 59 is an odd number.

Since the sum of the numbers on each of the 3 cards are equal, we can deduce that there are 2 even numbers and 1 odd number, or 1 even number and 2 odd numbers on the hidden sides of the card.

The only prime number which is even is 2. Therefore, there must be an odd number on each hidden sides of the number 44 and 38 and the number 2 is on the hidden side of the number 59.

59 + 2 = 61. Hence, the number on the hidden side of the number 44 is 61 - 44 = 17; the number on the hidden side of the number 38 is 61 - 38 = 23.

The average of the prime numbers is $\frac{2+17+23}{3}=\boxed{14}$