Final answer:
The answer involves interpreting the provided probability expressions and attempting to derive them using common probability rules. However, based on the information provided, a correct and complete derivation cannot be accomplished without further clarification of the terms and the context of the question.
Step-by-step explanation:
Understanding Probability Statements
The student's question relates to probability formulas and their derivations. However, there seems to be a confusion or typo in the expressions provided (e.g., p (ef c)). Assuming the student meant P(Ec | Fc) and P(Ec ∩ Fc), we can proceed with common probability rules to derive these expressions.
Part a: To prove P(Ec | Fc) = P(E) - P(E | F), we would typically need to start with the definition of conditional probability and manipulate the expressions accordingly. However, based on the information provided in the question, this derivation cannot be completed correctly.
Part b: For the expression P(Ec ∩ Fc) = 1 - P(E) - P(F) + P(E ∩ F), we use the fact that the probability of the union of two independent events E and F is equal to the sum of their individual probabilities minus their intersection probability, which is derived from the principle of inclusion-exclusion in probability theory.