91Ó°ÊÓ

In probability theory, understanding mutually exclusive events is crucial. These are events that simply cannot happen simultaneously. For example, think of rolling a dice. Rolling a 1 and a 6 at the same turn is mutually exclusive, since a single roll can only result in one number.

When events are mutually exclusive, the occurrence of one event prevents the occurrence of another. This characteristic simplifies probability calculations since the probability of either event happening is the sum of their individual probabilities. For events like these, the probability formula is straightforward:

• \[ P(X \cup Y) = P(X) + P(Y) \]

In our exercise, events \(X\) and \(Y\) are mutually exclusive. Therefore, you can just add \(P(X)\) and \(P(Y)\) to find \(P(X \cup Y)\). This results in a probability of 0.07 for either event \(X\) or \(Y\) occurring.

The complement rule is a powerful tool in probability that connects possibilities to certainty. It states that the probability of the complement of an event is equal to one minus the probability of the event itself. To put it simply, if you know the probability of something happening, you can easily find the probability of it not happening.

The mathematical expression for the complement rule is:

• \[ P( ext{not } A) = 1 - P(A) \]

In the context of our exercise, to determine the probability that neither \(X\) nor \(Y\) occur, we first find the probability that either \(X\) or \(Y\) occur using their sum since they are mutually exclusive. Then, applying the complement rule:

• \[ P( ext{neither } X ext{ nor } Y) = 1 - P(X \cup Y) = 0.93 \]

So, there is a 93% chance that neither event \(X\) nor event \(Y\) will occur.

Event probability measures how likely an occurrence or outcome will happen. In the framework of probability theory, it's essential to calculate and understand individual and combined probabilities. For a single event with its uncertainty captured in values between 0 and 1, the extremes represent impossibility and certainty, respectively.

For event \(X\) where \(P(X) = 0.05\), it signifies a 5% chance of happening. Similarly, for event \(Y\), \(P(Y) = 0.02\) translates to a 2% chance. Understanding these figures tells us about the likelihood of the events on their own.

To dive deeper, when considering more than one event, like those that are mutually exclusive, we use the addition of their probabilities to find the chance of either occurring:\[ P(X \cup Y) = 0.05 + 0.02 = 0.07 \].

Thus, the detailed comprehension of individual event probabilities helps in forming a base for more complex probability scenarios and solutions.