Eddy draws $6$ cards from a standard $52$-card deck. What is the probability that four of the cards that he draws have the same value?
1. **Calculate the total number of ways to draw 6 cards from a 52-card deck:**
   \[
   \binom{52}{6} = \frac{52!}{6!(52-6)!} = \frac{52!}{6! \cdot 46!}
   \]

2. **Determine the number of ways to get four cards of the same value:**
   - There are 13 different values (ranks) in a deck (2 through 10, Jack, Queen, King, Ace).
   - Choose one of these 13 values for the four cards:
     \[
     \binom{13}{1} = 13
     \]
   - For the chosen value, select 4 out of the 4 available cards:
     \[
     \binom{4}{4} = 1
     \]
   - For the remaining 2 cards, choose 2 cards from the remaining 48 cards (since 4 cards are already chosen):
     \[
     \binom{48}{2} = \frac{48!}{2!(48-2)!} = \frac{48!}{2! \cdot 46!}
     \]

3. **Calculate the number of valid hands:**
   \[
   13 \cdot \binom{48}{2} = 13 \cdot \frac{48!}{2! \cdot 46!} = 13 \cdot \frac{48 \cdot 47}{2} = 13 \cdot 1128 = 14664
   \]

4. **Calculate the probability:**
   \[
   \text{Probability} = \frac{\text{Number of valid hands}}{\text{Total number of hands}} = \frac{13 \cdot \binom{48}{2}}{\binom{52}{6}} = \frac{14664}{20358520}
   \]

5. **Simplify the fraction:**
   \[
   \frac{14664}{20358520} = \frac{3}{4165}
   \]

The final answer is \(\boxed{\frac{3}{4165}}\).