To solve this problem, you'll need to apply the formula for conditional probability and the multiplication rule. Here's a breakdown:

1. Probability of choosing an unsatisfactory piston from Machine 1:
   $P(\text{U}_1) = \frac{\text{Number of unsatisfactory pistons from Machine 1}}{\text{Total pistons from Machine 1}}$

2. Probability of choosing a satisfactory piston from Machine 2:
   $P(\text{S}_2) = \frac{\text{Number of satisfactory pistons from Machine 2}}{\text{Total pistons from Machine 2}}$

3. Multiplication Rule:
   If two events are independent, the probability of both occurring is the product of the individual probabilities. The final probability will be:
   $P(\text{U}_1 \cap \text{S}_2) = P(\text{U}_1) \times P(\text{S}_2)$

This approach considers that you're randomly choosing one piston from each machine and calculating the probability of picking an unsatisfactory piston from Machine 1 and a satisfactory one from Machine 2.

Would you like more details or need assistance with any step? Here are some follow-up questions:

1. How do you calculate conditional probabilities in general?
2. How can you interpret the multiplication rule in probability?
3. What other probability rules might be useful in this context?
4. Could this type of problem be approached differently using other concepts like Bayes' theorem?
5. How do you deal with dependent events in probability problems?

Tip: Always confirm whether events are independent or dependent before applying the multiplication rule.