Mikko takes a multiple choice test with ten questions. His only goal is to pass the test, and this requires seven points. A correct answer is worth one point, and answering wrong results in the deduction of one point. Mikko knows for sure that he knows the correct answer in the six first questions. For the rest, he estimates that he can give the correct answer to each problem with probability $p, 0 < p < 1$. How many questions Mikko should try?
1. **Define the problem and variables:**
   Mikko needs to pass a test with 10 questions, requiring at least 7 points to pass. He knows the correct answers to the first 6 questions. For the remaining 4 questions, he has a probability \( p \) of answering correctly and a probability \( 1-p \) of answering incorrectly. Each correct answer gives him 1 point, and each incorrect answer deducts 1 point.

2. **Set up the expected value equation:**
   Let \( x \) be the number of questions Mikko attempts out of the remaining 4 questions. The expected number of points he will get from these \( x \) questions is:
   \[
   E(\text{points}) = x \cdot p - x \cdot (1-p)
   \]
   Simplify the expression:
   \[
   E(\text{points}) = x \cdot p - x \cdot (1 - p) = x \cdot (2p - 1)
   \]

3. **Determine the required expected value:**
   Mikko already has 6 points from the first 6 questions. He needs at least 1 more point to reach 7 points. Therefore, we set up the inequality:
   \[
   x \cdot (2p - 1) \geq 1
   \]

4. **Solve for \( x \):**
   \[
   x \geq \frac{1}{2p - 1}
   \]

5. **Determine the least number of questions Mikko should try:**
   Since \( x \) must be an integer, we take the ceiling of the right-hand side:
   \[
   x \geq \lceil \frac{1}{2p - 1} \rceil
   \]

Conclusion:
Thus, the least number of questions Mikko should try is \( \lceil \frac{1}{2p - 1} \rceil \).

The final answer is \( \boxed{ \lceil \frac{1}{2p - 1} \rceil } \).