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Conditional probability is the measure of the likelihood of one event occurring with some relationship to one or more other events. When you're told that one event, say Event F, has occurred, this information can affect the probability of another event, say Event E, happening. The formula for conditional probability is given by:
\( P(E | F) = \frac{P(E \cap F)}{P(F)} \)
This formula states that the probability of E occurring given F has occurred is the ratio of the probability of both E and F occurring to the probability of F occurring. However, in our exercise, since E and F are mutually exclusive, they can't happen together.

Understanding the Implications

Knowing that Events E and F cannot occur at the same time is crucial. Hence, the conditional probability of E given F is simply 0 because the occurrence of F entirely rules out the occurrence of E. This is a key concept for tackling problems involving mutually exclusive events; by knowing the nature of the relationship between events, their probabilities can be effectively calculated.

The probability of an intersection refers to the likelihood of two events both taking place at the same time. For independent events, calculating this probability involves multiplying the probability of each event occurring separately. The general formula for the probability of the intersection of two events E and F is:
\( P(E \cap F) = P(E) \times P(F) \)
But what about when events are mutually exclusive, like in our textbook example? For mutually exclusive events, the intersection is empty because the events cannot occur together—hence, \(P(E \cap F)=0\).

Real-World Application

Understanding the concept of intersection is practical while dealing with real-life scenarios where events can or cannot happen simultaneously. For instance, drawing a heart and a club from a deck of cards in one pick is impossible—these are mutually exclusive events, just like E and F in our exercise.

Independent events are fundamentally different from mutually exclusive events. Two events are considered independent if the occurrence of one does not affect the probability of the occurrence of the other. One classic example is flipping a coin: whether it lands heads or tails on one flip has no bearing on the results of the next flip.
For independent events, the probability of E given F is simply the probability of E itself since F's occurrence does not influence E. This can be expressed as:
\( P(E | F) = P(E) \)
In the context of our original exercise, although it was mentioned that the occurrence of mutually exclusive events E and F is independent of each other, this isn't a reference to 'independent events' in the typical probabilistic sense. Instead, it means that knowing F occurred gives us no hope of E occurring; they don't influence one another because they simply cannot co-occur.