91Ó°ÊÓ

Probability is all about finding how likely an event is to occur. Sometimes, though, we are interested in finding out how likely something won't happen. That's when the Complementary Rule comes into play. It states simply that the probability of something not happening is equal to 1 minus the probability of it happening.

For example, if you want to know how likely it is that an event A will not occur, you calculate the complement:

• The probability of not A, denoted as \(P(A^{\prime})\), is \(1 - P(A)\).

In our original problem, we're looking for \(P((A \cup B)^{\prime})\), which means we're interested in finding the probability of not A or B happening. Using the Complementary Rule, we're just taking one minus the probability of A or B occurring. This makes it easy to find out probabilities for opposites, which often saves a lot of calculation.

The Union of Sets is like a big tent that covers all possibilities involved. In probability, the union of two or more events includes anything that happens in any of the events. If we're exploring the union of events A and B, written as \(A \cup B\), we're looking at whether A happens, B happens, or both happen.

To calculate the probability of this union, we use the formula that comes from combining their individual probabilities:

• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

By subtracting \(P(A \cap B)\), the overlap between the two events, we ensure that we don't double-count any event.
In the exercise, A and B were mutually exclusive, meaning \(P(A \cap B) = 0\). That made calculating the union straightforward, as there was no overlap to worry about.

Mutually Exclusive Events are scenarios where two events cannot happen at the same time. Think of it like having a coin that can land either heads or tails, but not both at once.

When events A and B are mutually exclusive, the probability of both A and B occurring is zero. This simplifies many calculations, as their intersection, \(P(A \cap B)\), is 0.

• In simpler terms, if \(P(A \cap B) = 0\), then A and B do not overlap.

In practical terms, this means that to find the probability of either A or B happening (but not both), you just add their individual probabilities: \(P(A) + P(B)\).

In our exercise, since events A and B were mutually exclusive, it became much easier to solve for \(P(A \cup B)\) because there was no chance of both occurring simultaneously. This is a frequent scenario in problems, making it essential to identify mutually exclusive events in probability.