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For two events [tex]\(X\)[/tex] and [tex]\(Y\)[/tex], [tex]\(P(X)=\frac{2}{3}\)[/tex], [tex]\(P(Y)=\frac{2}{5}\)[/tex], and [tex]\(P(X \cap Y)=\frac{1}{5}\)[/tex]. Find the probabilities.

1. [tex]\(P( Y \cap X ) = \frac{1}{5}\)[/tex]

2. [tex]\(P(Y) \cdot P(X) = \frac{4}{15}\)[/tex]



Answer :

GinnyAnswer
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]

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