Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Kat will win?
1. Let $K$ be the probability that Kat will win at some given instant when it is her turn.
2. We need to consider the sequence of events leading up to Kat's turn:
   - Gwen flips the coin first. If she gets heads, she wins, and the game ends.
   - If Gwen gets tails, Eli flips the coin. If Eli gets heads, he wins, and the game ends.
   - If Eli gets tails, it is now Kat's turn to flip the coin.
3. The probability that Gwen gets tails and Eli gets tails (so that it becomes Kat's turn) is:
   \[
   \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) = \frac{1}{4}
   \]
4. When it is Kat's turn, she has a $\frac{1}{2}$ probability of winning by flipping heads.
5. If Kat flips tails, the game resets to the initial state, but with the same probability structure. Therefore, the probability that Kat wins given that it is her turn can be expressed as:
   \[
   K = \frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{4} \cdot K\right)
   \]
6. Simplifying the equation:
   \[
   K = \frac{1}{2} + \frac{1}{8}K
   \]
7. To solve for $K$, we isolate $K$ on one side of the equation:
   \[
   K - \frac{1}{8}K = \frac{1}{2}
   \]
   \[
   \frac{7}{8}K = \frac{1}{2}
   \]
   \[
   K = \frac{1}{2} \cdot \frac{8}{7} = \frac{4}{7}
   \]
8. The probability that Kat wins on her first turn is:
   \[
   \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8}
   \]
9. The probability that Kat wins on subsequent turns is:
   \[
   \frac{1}{4} \cdot \frac{4}{7} = \frac{1}{7}
   \]

The final answer is $\boxed{\frac{1}{7}}$