To determine the probability that $E$ occurs before $F$, we can ignore
all the (independent) trials on which neither $E$ nor $F$ occurred,
that is, $(E\cup F)^c$ occurred, since we are going to repeat the
experiment until one of $E$ and $F$ does occur. So, look at the
trial of the experiment on which one of $E$ and $F$ has occurred
for the very first time.
We are given that on this trial, the event $E \cup F$ has occurred.
But, we don't yet know which of the two has occurred. So, given the
knowledge that $E \cup F$ has occurred, what is the conditional
probability that it was $E$ that occurred (and so $E$ occurred before $F$
since this is the first time we have seen either $E$ or $F$)?
$$P(E \mid (E \cup F)) = \frac{P(E(E \cup F))}{P(E \cup F)}
= \frac{P(E \cup EF)}{P(E) + P(F) - P(EF)}
= \frac{P(E)}{P(E)+P(F)}$$
since $P(EF) = P(\emptyset) = 0$.
Alternatively, let $G = (E\cup F)^c = E^c \cap F^c$ be the event that neither
$E$ nor $F$ occurs on a trial of the experiment. Note that
$P(G) = 1 - P(E) - P(F)$. Then, the event $E$ occurs
before $F$ if and only if one of the following compound events occurs:
$$
E, (G, E), (G, G, E), \ldots, (\underbrace{G, G, \ldots, G,}_{n-1} E), \ldots
$$
where $(\underbrace{G, G, \ldots, G,}_{n-1} E)$ means $n-1$ trials on which $G$
occurred and then $E$ occurred on the $n$-th trial. The desired probability
is thus
$$P(E ~\text{before}~ F) = P(E) + P(G)P(E) + [P(G)]^2P(E) + \cdots
= \frac{P(E)}{1 - P(G)} = \frac{P(E)}{P(E)+P(F)}.$$
Independence means that
$$P(A|C)=\frac{P(A\cap C)}{P(C)}=P(A)$$
$$\implies P(A\cap C)=P(A)P(C)$$
I.e, the probability of $A$ occurring wasn't affected by the prior occurrence of $C$. So the probability that $A$ and $C$ will occur (which is $P(A\cap C)$) is $$P(A)P(C)=0.2\cdot 0.5=0.1$$
Best Answer
"Mutually exclusive" and "independent" mean different things.
Two events are mutually exclusive if they can't both happen. For example, "my first name is Steve" and "my first name is Fred" are mutually exclusive. When events are mutually exclusive, you are allowed to add their probabilities to get the probability that one of them occurs.
Independent events are events where finding out about one doesn't change the probability of the other. Finding out that "it is raining" doesn't tell you anything about "my car is red" so those events are independent.
In the question the events "Dick hits the target", "Betty hits the target" and "Joe hits the target" are independent, but are not mutually exclusive. For instance, Betty and Dick could both hit the target. Since they are not mutually exclusive adding the probabilities will not give the correct answer.