Final answer:
The probability P(E|E^F) is undefined since E and F, which are mutually exclusive, can't occur at the same time, and P(EnF'|F) simplifies to P(E|F), which is 0 since E and F cannot happen simultaneously. Properties of independent and mutually exclusive events are numerically justified, such as P(U AND V) = 0 for mutually exclusive events U and V.
Step-by-step explanation:
Understanding Conditional Probability
To calculate P(E|E^F), which is the probability of event E given that the compound event E AND F has occurred, it's important to first understand what is meant by these symbols. However, since E and F are mutually exclusive (which means they cannot happen at the same time), the probability P(E|E^F) is undefined, as the event E^F (E intersect F) has a probability of 0. The definition of mutually exclusive events can be checked numerically: P(E AND F) is indeed 0.
Similarly, for P(EnF'|F), we want to calculate the probability of EnF' given F has occurred which simplifies to P(E|F), since E and F' (the complement of F) are non-overlapping by definition of a complement. Since E and F are mutually exclusive as given in the problem, P(E|F) is 0.
Properties of Independent and Mutually Exclusive Events
Independent events don't affect each other's probabilities, e.g., P(J) can be deduced using given probabilities of P(JK). Mutually exclusive events, on the other hand, cannot occur simultaneously, thus if we know P(U) and P(V), we know that P(U AND V) = 0 and P(U|V) = 0, since one occurring means the other cannot. Moreover, for any two events, mutual exclusivity is numerically justified if P(E AND F) = 0.