4. Partial Differential Equations

Try these first

1. 1.

   What is Conditional Probability?

   Hint

   Think about the probability of A in the context of B.

2. 2.

   If the probability of event B is 0.4, what does this imply about calculating P(A|B)?

   Hint

   Focus on the condition here.

3. 3.

   What is the formula for Conditional Probability?

   • P(A|B) = P(A ∩ B) / P(B)
   • P(A|B) = P(B) / P(A)
   • P(A|B) = P(A) + P(B)

   Hint

   Remember to consider what the condition is referring to.

4. 4.

   True or False: If A and B are independent, then P(A|B) equals P(A).

   • True
   • False

   Hint

   Think about what independence means.

5. 5.

   In a factory, 70% of the time a machine operates without errors. Consider a scenario where 90% of the errors are due to programming issues (P(E|P) = 0.9) and 10% are mechanical (P(E|M) = 0.1). If a machine fails, what is the probability that it was due to programming issues?

   Hint

   Break down the error probabilities into parts and apply.

6. 6.

   Two fair coins are flipped. What is the condition probability that at least one coin is heads, given that one of them is known to be heads?

   Hint

   Draw out the sample space considering the conditions.