91Ó°ÊÓ

In probability theory, when we talk about the union of events, we refer to the possibility that at least one of the events occurs. It's about combining both happenings.
For events \(A\) and \(B\), the union is represented as \(A \cup B\). It means the occurrence of event \(A\), event \(B\), or both.
The probability of the union of two events is calculated using the formula:

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

This formula accounts for the probabilities of both events and subtracts their intersection to avoid double-counting.
When solving problems involving unions, always consider the overlap between events, which is represented by their intersection.

The intersection of events in probability refers to all outcomes shared by two or more events. When events \(A\) and \(B\) occur together, it is called their intersection, denoted as \(A \cap B\).
Think of it as the common ground where both events meet.
Intersection probabilities are crucial when finding joint probabilities, by ensuring we only count those outcomes where both conditions are met.
In the context of our original exercise, the given \( P(A \cap B) = \frac{3}{10} \) represents the probability that both events \( A \) and \( B \) happen simultaneously.
Understanding intersections helps in solving more complex probability problems where multiple events are involved.

The probability formula is the backbone of calculating how likely an event is to occur. Key formulas include those for the union and intersection of events, which we used in our exercise.
For any events \(A\) and \(B\), with a known union and intersection, the primary formula applied is:

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

This formula ensures that we don't overcount the probability of events happening.
When something like \(P(B)\) is unknown, rearranging the original equation allows us to solve for it using the other probabilities given in the problem.
By substituting the known values and solving algebraically, we can find the missing probability. This method ensures we leverage every bit of available information to solve problems effectively.