To find the value of $P(Y > 2 \mid X > 1)$, we need to apply the concept of conditional probability. The conditional probability $P(Y > 2 \mid X > 1)$ represents the probability that $Y$ is greater than 2 given that $X$ is greater than 1.

The conditional probability formula is:

$P(Y > 2 \mid X > 1) = \frac{P(Y > 2 \text{ and } X > 1)}{P(X > 1)}$

To calculate this, we need:

1. $P(Y > 2 \text{ and } X > 1)$: This is the joint probability that both $Y > 2$ and $X > 1$ occur simultaneously.
2. $P(X > 1)$: This is the probability that $X > 1$.

Without additional information about the distribution of $X$ and $Y$, such as joint or marginal probability distributions, we cannot directly calculate these probabilities. However, if you can provide details on the distributions of $X$ and $Y$, such as whether they are independent or follow specific distributions (e.g., normal, exponential), we can proceed with calculating the required probabilities.

Would you like to provide more details, or should I explain further about general cases?

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Follow-Up Questions:

1. Are $X$ and $Y$ independent random variables?
2. Do $X$ and $Y$ follow any specific distributions (e.g., normal, exponential)?
3. Is there a known joint probability distribution for $X$ and $Y$?
4. Are there any given numerical values for $P(X > 1)$ and $P(Y > 2 \text{ and } X > 1)$?
5. Would you like an explanation of how to approach this problem if $X$ and $Y$ are independent?

Tip: For conditional probabilities, knowing if variables are independent can simplify calculations by allowing the use of the product rule for probabilities.