To determine whether the events [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are independent, we need to check whether the following condition holds true:
[tex]\[ P(X \text{ and } Y) = P(X) \times P(Y) \][/tex]
Given:
- [tex]\( P(X \text{ and } Y) = 0.2 \)[/tex]
- [tex]\( P(X \mid Y) = 0.4 \)[/tex]
- [tex]\( P(Y \mid X) = 0.5 \)[/tex]
### Step 1: Calculate [tex]\( P(X) \)[/tex]
We use the formula for conditional probability:
[tex]\[ P(X \mid Y) = \frac{P(X \text{ and } Y)}{P(Y)} \][/tex]
Rearranging for [tex]\( P(Y) \)[/tex]:
[tex]\[ P(Y) = \frac{P(X \text{ and } Y)}{P(X \mid Y)} \][/tex]
Plugging in the given values:
[tex]\[ P(Y) = \frac{0.2}{0.4} = 0.5 \][/tex]
### Step 2: Calculate [tex]\( P(Y) \)[/tex]
We use a similar formula for conditional probability:
[tex]\[ P(Y \mid X) = \frac{P(X \text{ and } Y)}{P(X)} \][/tex]
Rearranging for [tex]\( P(X) \)[/tex]:
[tex]\[ P(X) = \frac{P(X \text{ and } Y)}{P(Y \mid X)} \][/tex]
Plugging in the given values:
[tex]\[ P(X) = \frac{0.2}{0.5} = 0.4 \][/tex]
### Step 3: Calculate [tex]\( P(X) \times P(Y) \)[/tex]
Now we calculate the product [tex]\( P(X) \times P(Y) \)[/tex]:
[tex]\[ P(X) \times P(Y) = 0.4 \times 0.5 = 0.2 \][/tex]
### Step 4: Compare [tex]\( P(X \text{ and } Y) \)[/tex] with [tex]\( P(X) \times P(Y) \)[/tex]
We compare the values:
[tex]\[ P(X \text{ and } Y) = 0.2 \][/tex]
[tex]\[ P(X) \times P(Y) = 0.2 \][/tex]
Since [tex]\( P(X \text{ and } Y) = P(X) \times P(Y) \)[/tex], events [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are indeed independent.
Therefore, I agree with Hiki. [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are independent events.
