Certainly! Let's solve for the given probabilities step-by-step.
### Step 1: Understanding the Notations
- [tex]\( P(X) \)[/tex] is the probability of event [tex]\( X \)[/tex].
- [tex]\( P(Y) \)[/tex] is the probability of event [tex]\( Y \)[/tex].
- [tex]\( P(X \cap Y) \)[/tex] is the probability of both events [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] occurring simultaneously.
Given that:
- [tex]\( P(X) = \frac{2}{3} \)[/tex]
- [tex]\( P(Y) = \frac{2}{5} \)[/tex]
- [tex]\( P(X \cap Y) = \frac{1}{5} \)[/tex]
### Step 2: Finding [tex]\( P(Y \cap X) \)[/tex]
The probability of the intersection of events [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] is already given as [tex]\( P(X \cap Y) \)[/tex].
[tex]\[
P(Y \cap X) = P(X \cap Y) = \frac{1}{5}
\][/tex]
Hence,
[tex]\[
P(Y \cap X) = 0.2
\][/tex]
### Step 3: Calculating [tex]\( P(Y) \cdot P(X) \)[/tex]
To find [tex]\( P(Y) \cdot P(X) \)[/tex], we simply multiply the individual probabilities of [tex]\( Y \)[/tex] and [tex]\( X \)[/tex].
[tex]\[
P(Y) \cdot P(X) = \left(\frac{2}{5}\right) \cdot \left(\frac{2}{3}\right) = \frac{2 \times 2}{5 \times 3} = \frac{4}{15}
\][/tex]
### Final Results
1. [tex]\(\boxed{P(Y \cap X) = 0.2}\)[/tex]
2. [tex]\(\boxed{P(Y) \cdot P(X) = 0.26666666666666666}\)[/tex]
