The random variables $X, Y$ can each take a finite number of integer values. They are not necessarily independent. Express $P(\min(X,Y)=k)$ in terms of $p_1=P(X=k)$, $p_2=P(Y=k)$ and $p_3=P(\max(X,Y)=k)$.
To express \( P(\min(X,Y)=k) \) in terms of \( p_1 = P(X=k) \), \( p_2 = P(Y=k) \), and \( p_3 = P(\max(X,Y)=k) \), we can follow these steps:

1. **Identify the events:**
   \[
   \{X = k\} \cup \{Y = k\} = \{X = k, Y \leq k\} \cup \{X = k, Y \geq k\} \cup \{X \leq k, Y = k\} \cup \{X \geq k, Y = k\}
   \]
   This simplifies to:
   \[
   \{X = k\} \cup \{Y = k\} = \{X = k, Y \leq k\} \cup \{X \leq k, Y = k\} \cup \{X = k, Y \geq k\} \cup \{X \geq k, Y = k\}
   \]

2. **Combine the events:**
   \[
   \{X = k\} \cup \{Y = k\} = \{\min(X,Y) = k\} \cup \{\max(X,Y) = k\}
   \]

3. **Intersection of events:**
   \[
   \{X = k, Y = k\} = \{\min(X,Y) = k, \max(X,Y) = k\}
   \]

4. **Probability of the union of events:**
   \[
   P[\{X = k\} \cup \{Y = k\}] = P[\{\min(X,Y) = k\} \cup \{\max(X,Y) = k\}]
   \]

5. **Using the principle of inclusion-exclusion:**
   \[
   P[X = k] + P[Y = k] - P[X = k, Y = k] = P[\min(X,Y) = k] + P[\max(X,Y) = k] - P[\min(X,Y) = k, \max(X,Y) = k]
   \]

6. **Simplify using the given probabilities:**
   Since \( P[X = k, Y = k] = P[\min(X,Y) = k, \max(X,Y) = k] \), we have:
   \[
   P[X = k] + P[Y = k] = P[\min(X,Y) = k] + P[\max(X,Y) = k]
   \]

7. **Solve for \( P(\min(X,Y) = k) \):**
   \[
   P[\min(X,Y) = k] = P[X = k] + P[Y = k] - P[\max(X,Y) = k]
   \]
   Substituting the given probabilities:
   \[
   P[\min(X,Y) = k] = p_1 + p_2 - p_3
   \]

The final answer is \( \boxed{p_1 + p_2 - p_3} \).