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Mutually exclusive events are events that cannot happen at the same time. In simpler terms, if one event occurs, the other cannot. This means that there is no overlap between mutually exclusive events.
For instance, in an experiment with 10 outcomes, if you have two events, A and B, they are mutually exclusive if there are no common outcomes. For events to be mutually exclusive, their intersection must be empty, which is mathematically written as:\[A \cap B = \emptyset\]In our example, event A consists of the outcomes \(\{3,4,6,9\}\) and event B consists of \(\{1,2,5\}\). When we compare these two sets, there are no numbers that appear in both sets. Thus, events A and B are mutually exclusive. Understanding this concept is fundamental in probability theory as it helps determine how likely an event is to invoke change in another event's occurrence.

Independent events occur in such a way that the occurrence of one event does not affect the likelihood of another. In probability terms, two events A and B are independent if:\[P(A \cap B) = P(A) \times P(B)\]This implies that the joint probability of A and B equals the product of their individual probabilities. Importantly, an event's occurrence gives no information about the other.
In our specific case, we first need to calculate the individual probabilities of A and B. Event A has 4 outcomes out of 10, so \(P(A) = \frac{4}{10} = 0.4\). Event B has 3 outcomes, so \(P(B) = \frac{3}{10} = 0.3\). Their joint probability with no common elements is 0, which matches their product \(0.4 \times 0.3\) equals 0. Therefore, A and B are independent.
This may confuse some, as mutual exclusivity often implies dependence; however, when events have separate conditions, mutual exclusivity can coexist with independence under special circumstances like this one.

Complementary events are two events that account for all possible outcomes in a given scenario. The complement of an event A, written as \(A'\), includes everything not in A.
To find the probability of the complementary event, subtract the probability of the original event from 1:\[P(A') = 1 - P(A)\]In the given problem, the complement of event A includes all outcomes not contained in A. Since A consists of \(\{3,4,6,9\}\), the complement \(A'\) is \(\{1,2,5,7,8,10\}\). Similarly, for B \(=\{1,2,5\}\), the complement \(B'\) is \(\{3,4,6,7,8,9,10\}\).
The probability of \(A'\) is calculated as \(0.6\) because \(6\) of the \(10\) outcomes are in \(A'\). For \(B'\), with \(7\) outcomes, the probability is \(0.7\). Complementary events ensure that we've accounted for every possibility within the experiment's scope.