To determine whether the events [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are mutually exclusive, let’s review some key concepts and follow the logical steps.
### Key Concepts:
1. Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time. When two events are mutually exclusive, the probability of their intersection is zero; that is, [tex]\( P(X \cap Y) = 0 \)[/tex].
2. Addition Rule for Mutually Exclusive Events: If events [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are mutually exclusive, then the probability of their union is the sum of their individual probabilities:
[tex]\[
P(X \cup Y) = P(X) + P(Y)
\][/tex]
### Given Information:
- [tex]\( P(Y) = 0.25 \)[/tex]
- [tex]\( P(X) = 0.2 \)[/tex]
For [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] to be mutually exclusive, the following condition needs to be satisfied:
[tex]\[
P(X \cap Y) = 0
\][/tex]
Given that they are mutually exclusive, we use the addition rule:
[tex]\[
P(X \cup Y) = P(X) + P(Y)
\][/tex]
Substitute the known probabilities:
[tex]\[
P(X \cup Y) = 0.2 + 0.25 = 0.45
\][/tex]
Since the sum of [tex]\( P(X) \)[/tex] and [tex]\( P(Y) \)[/tex] is [tex]\( 0.45 \)[/tex], which is less than or equal to [tex]\( 1 \)[/tex], the condition for the probabilities of mutually exclusive events is satisfied.
### Conclusion:
Given that the sum of [tex]\( P(X) \)[/tex] and [tex]\( P(Y) \)[/tex] is less than or equal to 1, the events [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are indeed mutually exclusive. Therefore, the correct answer is:
True
