The joint probabilities are given in the table. The value of P(Y<=2) is3/161/163/323/64

To find $P(Y \leq 2)$ using the joint probabilities from the provided table, we can outline our approach as follows:

1. Break Down the Problem

Identify values for $Y$ that are less than or equal to $2$. Exclude any probability mass for values greater than $2$.

2. Relevant Concepts

Joint probabilities typically are given in a probability distribution table. We need to sum the probabilities corresponding to the values of $Y$ that satisfy $Y \leq 2$.

3. Analysis and Detail

Assuming we have access to the specific joint probability table, we would locate and sum the probabilities $P(Y = 0)$, $P(Y = 1)$, and $P(Y = 2)$: $P(Y \leq 2) = P(Y = 0) + P(Y = 1) + P(Y = 2)$

4. Verify and Summarize

After calculating the sum, verify that it indeed represents the cumulative probability for $Y$ values up to $2$.

Final Answer

The value of $P(Y \leq 2)$ based on your selections and the joint probability table is one of the given options: $\frac{3}{16}, \frac{1}{16}, \frac{3}{32}, \frac{3}{64}$.

Without the actual table data, I cannot compute the exact total. Please provide the joint probability values for specific calculations.